Graph of groups decompositions of graph braid groups

We provide an explicit construction that allows one to easily decompose a graph braid group as a graph of groups. This allows us to compute the braid groups of a wide range of graphs, as well as providing two general criteria for a graph braid group to split as a non-trivial free product, answering two questions of Genevois. We also use this to distinguish certain right-angled Artin groups and graph braid groups. Additionally, we provide an explicit example of a graph braid group that is relatively hyperbolic, but is not hyperbolic relative to braid groups of proper subgraphs. This answers another question of Genevois in the negative.


Introduction
Given a topological space X, one can construct the configuration space C top n pXq of n particles on X by taking the direct product of n copies of X and removing the diagonal consisting of all tuples where two coordinates coincide. Informally, this space tracks the movement of the particles through X; removing the diagonal ensures the particles do not collide. One then obtains the unordered configuration space U C top n pXq by taking the quotient by the action of the symmetric group by permutation of the coordinates of C top n pXq. Finally, the braid group B n pX, Sq is defined to be the fundamental group of U C top n pXq with base Date: September 9, 2022. point S (in general we shall assume X to be connected and drop the base point from our notation). The fundamental group of C top n pXq is called the pure braid group P B n pXq. This description of braid groups is originally due to Fox and Neuwirth [FN62].
Classically, the space X is taken to be a disc. However, one may also study braid groups of other spaces. Here, we study the case where X is a finite graph Γ. These so-called graph braid groups were first developed by Abrams [Abr00], who showed that B n pΓq can be expressed as the fundamental group of a non-positively curved cube complex U C n pΓq; this was also shown independently by Świątkowski [Świ01, Proposition 2.3.1]. Results of Crisp-Wiest, and later Genevois, show that these cube complexes are in fact special [CW04,Gen21b], in the sense of Haglund and Wise [HW08].
As fundamental groups of special cube complexes, graph braid groups B n pΓq embed in right-angled Artin groups [HW08, Theorem 1.1]. In fact, Sabalka also shows that all rightangled Artin groups embed in graph braid groups [Sab07, Theorem 1.1]. It is therefore natural to ask when a graph braid group is isomorphic to a right-angled Artin group. This question was first studied by Connolly and Doig in the case where Γ is a linear tree [CD14], and later by Kim, Ko, and Park, who show that for n ě 5, B n pΓq is isomorphic to a rightangled Artin group if and only if Γ does not contain a subgraph homeomorphic to the letter "A" or a certain tree [KKP12,Theorems A,B]. We strengthen this theorem by showing that if Γ does not contain a subgraph homeomorphic to the letter "A", then B n pΓq must split as a non-trivial free product. Thus, for n ě 5, B n pΓq is never isomorphic to a right-angled Artin group with connected defining graph containing at least two vertices.
Theorem A (Theorem 4.7, [KKP12,Theorem B]). Let Γ and Π be finite connected graphs with at least two vertices and let n ě 5. Then B n pΓq is not isomorphic to the right-angled Artin group A Π .
In fact, we prove two much more general criteria for a graph braid group to split as a non-trivial free product. In the theorem below, a flower graph is a graph obtained by gluing cycles and segments along a single central vertex.
Theorem B (Lemmas 4.1 and 4.2). Let n ě 2 and let Γ be a finite graph whose edges are sufficiently subdivided. Suppose one of the following holds: ‚ Γ is obtained by gluing a non-segment flower graph Φ to a connected non-segment graph Ω along a vertex v, where v is either the central vertex of Φ or a vertex of Φ of valence 1; ‚ Γ contains an edge e such that Γ e is connected but Γ e is disconnected, and one of the connected components of Γ e is a segment. Then B n pΓq -H˚Z for some non-trivial subgroup H of B n pΓq.
It would be interesting to know if the converse is true; the author is not aware of any graph braid groups that split as free products but do not satisfy either of the above criteria.
Question C. Are there any graph braid groups that split as non-trivial free products but do not satisfy the hypotheses of Theorem B?
In light of Theorem A, it is natural to ask in which ways the behaviour of graph braid groups is similar to that of right-angled Artin groups and in which ways it differs. One approach to this is to study non-positive curvature properties of graph braid groups. For example, it is known that a right-angled Artin group A Π is hyperbolic if and only if Π contains no edges, A Π is relatively hyperbolic if and only if Π is disconnected, and A Π is toral relatively hyperbolic if and only if Π is disconnected and every connected component is a complete graph. Furthermore, if A Π is relatively hyperbolic, then its peripheral subgroups are rightangled Artin groups whose defining graphs are connected components of Π. One can obtain similar, albeit more complicated, graphical characterisations of (relative) hyperbolicity in right-angled Coxeter groups W Π ; see [Mou88,Lev19]. Once again, the peripheral subgroups appear as right-angled Coxeter groups whose defining graphs are subgraphs of Π.
Genevois shows that in the case of hyperbolicity, graph braid groups admit a graphical characterisation [Gen21b, Theorem 1.1]: for connected Γ, B 2 pΓq is hyperbolic if and only if Γ does not contain two disjoint cycles; B 3 pΓq is hyperbolic if and only if Γ is a tree, a sun graph, a flower graph, or a pulsar graph; and for n ě 4, B n pΓq is hyperbolic if and only if Γ is a flower graph. However, one shortcoming of this theorem is that it introduced classes of graphs whose braid groups were unknown: sun graphs and pulsar graphs (braid groups on flower graphs are known to be free by [Gen21b,Lemma 4.6], while braid groups on trees are known to be free for n " 3 by [FS05, Theorems 2.5, 4.3]). By applying Theorem B, we are able to (partially) answer a question of Genevois [Gen21b,Question 5.3] and thus provide a more complete algebraic characterisation of hyperbolicity. The only exception is when Γ is a generalised theta graph, which proves resistant to computation. Here, a generalised theta graph Θ m is a graph obtained by gluing m cycles along a non-trivial segment.
Theorem D (Theorem 4.3, [Gen21b, Theorem 1.1]). Let Γ be a finite connected graph that is not homeomorphic to Θ m for any m ě 0. The braid group B 3 pΓq is hyperbolic only if B 3 pΓq -H˚Z for some group H.
Genevois also provides a graphical characterisation of toral relative hyperbolicity [Gen21b,Theorem 4.20]. Again, this theorem introduced several classes of graphs for which the braid groups were unknown. In the case of n " 4, this is a finite collection of graphs: the graphs homeomorphic to the letters "H", "A", and "θ". We are able to precisely compute the braid groups of these graphs, completing the algebraic characterisation of toral relative hyperbolicity for n " 4 and answering a question of Genevois [Gen21b,Question 5.6].
Theorem E (Theorem 3.15, [Gen21b,Theorem 4.20]). Let Γ be a finite connected graph. The braid group B 4 pΓq is toral relatively hyperbolic only if it is either a free group or isomorphic to F 10˚Z 2 or F 5˚Z 2 or an HNN extension of Z˚Z 2 .
It is much more difficult to characterise relative hyperbolicity in general for graph braid groups. Indeed, we show that in some sense it is impossible to obtain a graphical characterisation of the form that exists for right-angled Artin and Coxeter groups, by providing an example of a graph braid group that is relatively hyperbolic but is not hyperbolic relative to any braid groups of proper subgraphs. This answers a question of Genevois in the negative [Gen21b, follow-up to Question 5.7].
Theorem F (Theorem 4.4). There exists a graph braid group B n pΓq that is hyperbolic relative to a thick, proper subgroup P that is not contained in any graph braid group of the form B k pΛq for k ď n and Λ Ĺ Γ. Γ X " U C 2 pΓq Figure 1. B 2 pΓq is isomorphic to π 1 pXq, which is hyperbolic relative to the subgroup P generated by the fundamental groups of the three shaded tori.
Note that the peripheral subgroup P in the above example is precisely the group constructed by Croke and Kleiner in [CK00, Section 3]. In particular, it is isomorphic to the right-angled Artin group A Π where Π is a segment of length 3. This theorem indicates that non-relatively hyperbolic behaviour cannot be localised to specific regions of the graph Γ. Instead, non-relative hyperbolicity is in some sense a property intrinsic to the special cube complex structure.
Theorems A, B, D, E, and F are all proved using a technical result on graph of groups decompositions of graph braid groups, which we believe to be of independent interest. Graph of groups decompositions were first considered for pure graph braid groups P B n pΓq by Abrams [Abr00] and Ghrist [Ghr01], and more recently Genevois produced a limited result of this flavour for B n pΓq [Gen21b, Proposition 4.6]. In this paper, we use the structure of U C n pΓq as a special cube complex to produce a general construction that allows one to explicitly compute graph of groups decompositions of graph braid groups. In particular, the vertex groups and edge groups are braid groups on proper subgraphs. By iterating this procedure, one is therefore able to express a graph braid group as a combination of simpler, known graph braid groups.
Theorem G (Theorem 3.5). Let Γ be a finite connected graph whose edges are sufficiently subdivided and let e 1 , . . . , e m be distinct edges of Γ sharing a common vertex. The graph braid group B n pΓq decomposes as a graph of groups pG, Λq, where: ‚ V pΛq is the collection of connected components K of U C n pΓ pe 1 Y¨¨¨Ye m qq; ‚ EpΛq is the collection of hyperplanes H of U C n pΓq labelled by some e i , where H joins K and L if it has one combinatorial hyperplane in K and another in L; ‚ for each K P V pΛq, we have G K " B n pΓ pe 1 Y¨¨¨Ye m q, S K q for some S K P K; ‚ for each H i P EpΛq labelled by e i , we have G H i " B n´1 pΓ e i , S H i X pΓ e i qq, for some S H i in one of the combinatorial hyperplanes Hȋ ; ‚ for each edge H P EpΛq joining vertices K, L P V pΛq, the monomorphisms φH are induced by the inclusion maps of the combinatorial hyperplanes H˘into K and L.
By selecting the edges e 1 , . . . , e m carefully, one may often be able to arrange for the edge groups of pG, Λq to be trivial. This results in a wide range of graph braid groups that split as non-trivial free products, as shown in Theorem B.
This construction also aids in the computation of specific graph braid groups, especially when combined with Propositions 3.7 and 3.8, in which we give combinatorial criteria for adjacency of two vertices of Λ, as well as providing a way of counting how many edges connect each pair of vertices of Λ. Traditionally, graph braid group computations are performed by using Farley and Sabalka's discrete Morse theory to find explicit presentations [FS05,FS12], however in practice these presentations are often highly complex, difficult to compute, and are obfuscated by redundant generators and relators. We believe the graph of groups approach to be both easier to apply and more powerful in many situations, as evidenced by our theorems and the numerous examples we are able to compute in Section 3.1.
Outline of the paper. We begin with some background on the geometry of cube complexes in Section 2.1, in particular introducing the notion of a special cube complex. We then introduce graph braid groups in Section 2.2, showing that they are fundamental groups of special cube complexes and providing some important foundational results. Section 2.3 then introduces the necessary material on graphs of groups.
Our main technical theorem, Theorem G, is proved in Section 3 (Theorem 3.5). We then apply this theorem to compute a number of specific examples of graph braid groups in Section 3.1, including braid groups of radial trees and graphs homeomorphic to the letters "H", "A", "θ", and "Q". These examples allow us to prove Theorem E (Theorem 3.15).
Section 4 deals with more general applications of Theorem G. In Section 4.1 we prove Theorem B, providing two general criteria for a graph braid group to decompose as a nontrivial free product (Lemmas 4.1 and 4.2). In Section 4.2, we prove Theorem F (Theorem 4.4), showing that there exist relatively hyperbolic graph braid groups that are not hyperbolic relative to any braid group of a subgraph. We then prove Theorem A (Theorem 4.7) in Section 4.3, showing that braid groups on large numbers of particles cannot be isomorphic to right-angled Artin groups with connected defining graphs.
We conclude by posing some open questions in Section 5. Definition 2.1 (Cube complex). Let n ě 0. An n-cube is a Euclidean cube r´1 2 , 1 2 s n . A face of a cube is a subcomplex obtained by restricting one or more of the coordinates to˘1 2 . A cube complex is a CW complex where each cell is a cube and the attaching maps are given by isometries along faces.
We will often refer to the 0-cubes of a cube complex X as vertices, the 1-cubes as edges, and the 2-cubes as squares.
Definition 2.2 (Non-positively curved, CAT(0)). Let X be a cube complex. The link linkpvq of a vertex v of X is a simplicial complex defined as follows.
‚ The vertices of linkpvq are the edges of X that are incident at v. ‚ n vertices of linkpvq span an n-simplex if the corresponding edges of X are faces of a common cube. The complex linkpvq is said to be flag if n vertices v 1 , . . . , v n of linkpvq span an n-simplex if and only if v i and v j are connected by an edge for all i ‰ j. A cube complex X is nonpositively curved if the link of each vertex of X is flag and contains no bigons (that is, no loops consisting of two edges). A cube complex X is CAT(0) if it is non-positively curved and simply connected.
The link condition tells us that a non-positively curved cube complex X is determined by its 1-skeleton X p1q . In general, we shall therefore work in this 1-skeleton, where we use the graph metric, denoted d X .
We study cube complexes from the geometric point of view of hyperplanes.
Definition 2.3 (Mid-cube, hyperplane, combinatorial hyperplane, carrier). Let X be a cube complex. A mid-cube of a cube C -r´1 2 , 1 2 s n of X is obtained by restricting one of the coordinates of C to 0. Each mid-cube has two isometric associated faces of C, obtained by restricting this coordinate to˘1 2 instead of 0. A hyperplane H of X is a maximal connected union of mid-cubes. A combinatorial hyperplane associated to H is a maximal connected union of faces associated to mid-cubes of H. The closed (resp. open) carrier of H is the union of all closed (resp. open) cubes of X which contain mid-cubes of H.
Convention 2.4. We will almost always be using the closed version of the carrier of a hyperplane H, therefore we will refer to this version as simply the 'carrier' of H.
A result of Chepoi tells us that carriers and combinatorial hyperplanes form convex subcomplexes of a CAT(0) cube complex [Che00, Proposition 6.6]. In the greater generality of non-positively curved cube complexes, this is not always true, however carriers and combinatorial hyperplanes are still locally convex, in the following sense.
Definition 2.5 (Local isometry, locally convex). A map φ : Y Ñ X between non-positively curved cube complexes is a local isometry if it is a local injection and for each vertex y P Y p0q and each pair of vertices u, v P Y p0q adjacent to y, if φpuq, φpyq, φpvq form a corner of a 2-cube in X, then u, y, v form a corner of a 2-cube in Y . A subcomplex of a non-positively curved cube complex X is locally convex if it embeds by a local isometry.
Proposition 2.6. Hyperplane carriers and combinatorial hyperplanes in a non-positively curved cube complex are locally convex.
Proof. Let X be a non-positively curved cube complex, let Y be a subcomplex of X that is either a hyperplane carrier or a combinatorial hyperplane, and let φ : Y Ñ X be the natural embedding of Y as a subcomplex of X. Note that the universal coverX is a CAT(0) cube complex, therefore hyperplane carriers and combinatorial hyperplanes inX are convex subcomplexes [Che00, Proposition 6.6].
Let y P Y p0q and let u, v P Y p0q be vertices adjacent to y such that φpuq, φpyq, φpvq form a corner of a 2-cube C Ď X. Lift Y to a subcomplexỸ ofX, letũ,ỹ,ṽ be the corresponding lifts of u, y, v, let ψ :Ỹ ÑX be the natural embedding ofỸ as a subcomplex ofX, and lift C to a 2-cubeC ĎX so that ψpũq, ψpỹq, ψpṽq form a corner ofC. By convexity ofỸ inX, u,ỹ,ṽ must also form a corner of a 2-cube inỸ . Projecting back down to Y , we therefore see that u, y, v form a corner of a 2-cube in Y , hence Y is locally convex in X.
The combinatorial hyperplanes associated to a given hyperplane are also 'parallel', in the following sense.
Definition 2.7 (Parallelism). Let H be a hyperplane of a cube complex X. We say that H is dual to an edge E of X if H contains a mid-cube which intersects E. We say that H crosses a subcomplex Y of X if there exists some edge E of Y that is dual to H. We say that H crosses another hyperplane H 1 if it crosses a combinatorial hyperplane associated to H 1 . Two subcomplexes Y, Y 1 of X are parallel if each hyperplane H of X crosses Y if and only if it crosses Y 1 .
Parallelism defines an equivalence relation on the edges of a cube complex X. In particular, two edges are in the same equivalence class (or parallelism class) if and only if they are dual to the same hyperplane. Therefore, one may instead consider hyperplanes of X to be parallelism classes of edges of X. One may also define an orientation Ý Ñ H on a hyperplane H by taking an equivalence class of oriented edges. In this case, we say that Ý Ñ H is dual to the oriented edges in this class.
Definition 2.8 (Osculation). We say H directly self-osculates if there exist two oriented edges Ý Ñ E 1 , Ý Ñ E 2 dual to Ý Ñ H that have the same initial or terminal vertex but do not span a square. We say two hyperplanes H 1 , H 2 inter-osculate if they intersect and there exist dual edges E 1 , E 2 of H 1 , H 2 , respectively, such that E 1 and E 2 share a common endpoint but do not span a square. See Figure 2.
Definition 2.9 (Special). A non-positively curved cube complex X is said to be special if its hyperplanes satisfy the following properties. (1) Hyperplanes are two-sided; that is, the open carrier of a hyperplane H is homeomorphic to Hˆp´1 2 , 1 2 q.
(2) Hyperplanes of X do not self-intersect.
Notation 2.10. Two-sidedness implies combinatorial hyperplanes associated to H are homeomorphic to Hˆt˘1 2 u. We denote the combinatorial hyperplane corresponding to Hˆt´1 2 u by H´and the combinatorial hyperplane corresponding to Hˆt 1 2 u by H`. 2.2. Graph braid groups. Consider a finite collection of particles lying on a finite metric graph Γ. The configuration space of these particles on Γ is the collection of all possible ways the particles can be arranged on the graph with no two particles occupying the same location. As we move through the configuration space, the particles move along Γ, without colliding. If we do not distinguish between each of the different particles, we call this an unordered configuration space. A graph braid group is the fundamental group of an unordered configuration space. More precisely: Definition 2.11 (Graph braid group). Let Γ be a finite graph, and let n be a positive integer. The topological configuration space C top n pΓq is defined as n pΓq is then defined as U C top n pΓq " C top n pΓq{S n , where the symmetric group S n acts on C top n pΓq by permuting its coordinates. We define the graph braid group B n pΓ, Sq as B n pΓ, Sq " π 1 pU C top n pΓq, Sq, where S P U C top n pΓq is a fixed base point. The base point S in our definition represents an initial configuration of the particles on the graph Γ. As the particles are unordered, they may always be moved along Γ into any other desired initial configuration, so long as the correct number of particles are present in each connected component of Γ. In particular, if Γ is connected, then the graph braid group B n pΓ, Sq is independent of the choice of base point, and may therefore be denoted simply B n pΓq.
Note that in some sense, the space U C top n pΓq is almost a cube complex. Indeed, Γ n is a cube complex, but removing the diagonal breaks the structure of some of its cubes. By expanding the diagonal slightly, we are able to fix this by ensuring that we are always removing whole cubes.
Definition 2.12 (Combinatorial configuration space). Let Γ be a finite graph, and let n be a positive integer. For each x P Γ, the carrier cpxq of x is the lowest dimensional simplex of Γ containing x. The combinatorial configuration space C n pΓq is defined as The reduced graph braid group RB n pΓ, Sq is defined as Removing this new version of the diagonal tells us that two particles cannot occupy the same edge of Γ. This effectively discretises the motion of the particles to jumps between vertices, as each particle must fully traverse an edge before another particle may enter.
Observe that C n pΓq is the union of all products cpx 1 qˆ¨¨¨ˆcpx n q satisfying cpx i qXcpx j q " H for all i ‰ j. Since the carrier is always a vertex or a closed edge, this defines an ndimensional cube complex, and moreover C n pΓq is compact with finitely many hyperplanes, as Γ is a finite graph. It follows that U C n pΓq is also a compact cube complex with finitely many hyperplanes. Indeed, we have the following useful description of the cube complex structure, due to Genevois.
Construction 2.13 ([Gen21b, Section 3]). The cube complex U C n pΓq may be constructed as follows.
‚ The vertices of U C n pΓq are the subsets S of V pΓq with size |S| " n. ‚ Two vertices S 1 and S 2 of U C n pΓq are connected by an edge if their symmetric difference S 1 S 2 is a pair of adjacent vertices of Γ. We therefore label each edge E of U C n pΓq with a closed edge e of Γ. ‚ A collection of m edges of U C n pΓq with a common endpoint span an m-cube if their labels are pairwise disjoint.
In the case n " 2, we can use this procedure to give a simple algorithm for drawing the cube complex U C 2 pΓq.
Algorithm 2.14. Let Γ be a finite graph with m vertices, and let n be a positive integer.
(3) Remove the diagonal and the upper triangle. The remaining lower triangle is the 0-skeleton U C 2 pΓq p0q , with the vertex w ij at the pi, jq coordinate corresponding to the configuration of one particle at vertex v i and one particle at vertex v j . (4) Add an edge between w ij and w kl if either i " k and tv j , v l u P EpΓq, or j " l and tv i , v k u P EpΓq, or i " l and tv j , v k u P EpΓq, or j " k and tv i , v l u P EpΓq. Label the edge between w ij and w kl with the corresponding edge of EpΓq. (5) For each vertex w ij and every pair of edges adjacent to w ij labelled by tv i , v k u and tv j , v l u for some k ‰ l, add a 2-cube whose 1-skeleton is the 4-cycle w ij , w kj , w kl , w il .
An example is given in Figure 3 below. Note, the darker shaded regions are the tori arising from products of disjoint 3-cycles in ∆.
1 2 3 4 5 6 7 8 9  Abrams showed that if Γ has more than n vertices, then U C n pΓq is connected if and only if Γ is connected.
Theorem 2.15 ([Abr00, Theorem 2.6]). Let Γ be a finite graph. If Γ has more than n vertices, then U C n pΓq is connected if and only if Γ is connected.
By analysing Abrams' proof, we are able to extract additional information regarding how connected components of Γ give rise to connected components of U C n pΓq.
Corollary 2.16. Let Γ be a finite graph with k connected components. If each component of Γ has at least n vertices, then U C n pΓq has`n`k´1 k´1˘c onnected components, corresponding to the number of ways to partition the n particles into the k connected components of Γ.
Proof. Abrams' proof of Theorem 2.15 proceeds by showing that if x, y P U C n pΓq are two configurations where n´1 of the particles are in the same locations in x and y, and the remaining particle lies in different components of Γ for x and y, then there is no path in U C n pΓq connecting x and y. Conversely, Abrams shows that if the remaining particle lies in the same component of Γ for x and y, then there is a path in U C n pΓq connecting x and y. By concatenating such paths, we see that two arbitrary configurations x, y P U C n pΓq are connected by a path if and only if x and y have the same number of particles in each connected component of Γ. Thus, the number of connected components of U C n pΓq is precisely the number of ways the n particles can be partitioned into the k components of Γ. Using the stars and bars technique, we therefore see that U C n pΓq has`n`k´1 k´1˘c onnected components.
Furthermore, Abrams showed that if we subdivide edges of Γ sufficiently (i.e. add 2-valent vertices to the middle of edges) to give a new graph Γ 1 , then U C top n pΓ 1 q deformation retracts onto U C n pΓ 1 q [Abr00, Theorem 2.1]. As U C top n pΓ 1 q does not distinguish vertices from other points on the graph, we have U C top n pΓ 1 q = U C top n pΓq, implying that B n pΓ, Sq -RB n pΓ 1 , Sq. This allows us to consider B n pΓ, Sq as the fundamental group of the cube complex U C n pΓ 1 q.
Prue and Scrimshaw later improved upon the constants in Abrams' result to give the following theorem [PS14, Theorem 3.2].
Theorem 2.17 ([Abr00], [PS14]). Let n P N and let Γ be a finite graph with at least n vertices. The unordered topological configuration space U C top n pΓq deformation retracts onto the unordered combinatorial configuration space U C n pΓq (and in particular RB n pΓq -B n pΓq) if the following conditions hold.
(1) Every path between distinct vertices of Γ of valence ‰ 2 has length at least n´1.
(2) Every homotopically essential loop in Γ has length at least n`1.
Note that Prue and Scrimshaw's version of the theorem only deals with the case where Γ is connected. However, the disconnected case follows easily by deformation retracting each connected component of U C n pΓq, noting that each component can be expressed as a product of cube complexes U C k pΛq, where k ď n and Λ is a connected component of Γ. In fact, we have the following more general result about configuration spaces of disconnected graphs, due to Genevois.
Lemma 2.18 ([Gen21b, Lemma 3.5]). Let n ą 1, let Γ be a finite graph, and suppose Moreover, if S P U C n pΓq has k particles in Γ 1 and n´k particles in Γ 2 , then where S X Γ 1 (resp. S X Γ 2 ) denotes the configuration of the k (resp. n´k) particles of S lying in Γ 1 (resp. Γ 2 ).
Remark 2.19. Note, Genevois states this result in terms of B n pΓ, Sq rather than RB n pΓ, Sq. However, the proof proceeds by showing it is true for RB n pΓ, Sq and then applying Theorem 2.17 after subdividing edges of Γ sufficiently. Indeed, this applies to all results of Genevois cited in this section.
Another foundational result of Abrams states that the cube complex U C n pΓq is nonpositively curved [Abr00, Theorem 3.10]. Furthermore, Genevois proved that U C n pΓq admits a special colouring [Gen21b, Proposition 3.7]. We shall omit the details of his theory of special colourings, and direct the reader to [Gen21a] for further details. The key result is that a cube complex X admits a special colouring if and only if there exists a special cube complex Y such that Y p2q " X p2q [Gen21a, Lemma 3.2]. Furthermore, Genevois constructs Y by taking X p2q and inductively attaching m-cubes C whenever a copy of C pm´1q is present in the complex, for m ě 3; this ensures non-positive curvature of Y . Since in our case X " U C n pΓq is already non-positively curved, this means Y " X. Thus, B n pΓq is the fundamental group of the special cube complex U C n pΓ 1 q. Crisp and Wiest also showed this indirectly by proving that every graph braid group embeds in a right-angled Artin group [CW04, Theorem 2].
In summary, we have the following result.
Corollary 2.20 (Graph braid groups are special; [Abr00, CW04, Gen21b, Gen21a]). Let n ą 1 and let Γ be a finite, connected graph. Then B n pΓq -RB n pΓ 1 q, where Γ 1 is obtained from Γ by subdividing edges. In particular, B n pΓq is the fundamental group of the connected compact special cube complex U C n pΓ 1 q.
Furthermore, Genevois provides a useful combinatorial criterion for detecting flats in graph braid groups. This is summarised by the following two lemmas.
Lemma 2.21 (Subgraphs induce subgroups [Gen21b, Proposition 3.10]). Let Γ be a finite, connected graph and let n ě 1. If Λ is a proper subgraph of Γ, then RB m pΛ, Sq embeds as a subgroup of RB n pΓq for all m ď n and all base points S P U C m pΛq.
Lemma 2.22 (Criterion for infinite diameter [Gen21b, Lemma 4.3]). Let Γ be a finite graph, let n ě 1, and let S be a vertex of U C n pΓq, so that S is a subset of V pΓq with size |S| " n. Then RB n pΓ, Sq has infinite diameter if and only if one of the following holds; otherwise, RB n pΓ, Sq is trivial.
‚ n " 1 and the connected component of Γ containing S has a cycle subgraph; ‚ n ě 2 and either Γ has a connected component whose intersection with S has cardinality at least 1 and which contains a cycle subgraph, or Γ has a connected component whose intersection with S has cardinality at least 2 and which contains a vertex of valence at least 3.
Theorem 2.23. Let Γ be a finite connected graph and let n ě 1. The reduced graph braid group RB n pΓq contains a Z 2 subgroup if and only if one of the following holds.
‚ n " 2 and Γ contains two disjoint cycle subgraphs; ‚ n " 3 and Γ contains either two disjoint cycle subgraphs or a cycle and a vertex of valence at least 3 disjoint from it; ‚ n ě 4 and Γ contains either two disjoint cycle subgraphs, or a cycle and a vertex of valence at least 3 disjoint from it, or two vertices of valence at least 3.
One may also produce similar theorems characterising Z m subgroups for any m ě 1.
2.3. Graphs of groups. A graph of groups is a system pG, Λq consisting of: ‚ an oriented connected graph Λ; ‚ a group G v for each vertex v P V pΛq (called a vertex group) and a group G e for each edge e P EpΛq (called an edge group); ‚ monomorphisms φé : G e Ñ G opeq and φè : G e Ñ G tpeq , where opeq and tpeq denote the initial and terminal vertices of e, respectively.
We recall two ways of defining a graph of groups decomposition of a group G.
Definition 2.24 (Fundamental group of pG, Λq). Let pG, Λq be a graph of groups and let T be a spanning tree for Λ. The fundamental group of pG, Λq based at T , denoted π 1 pG, Λ; T q, is the quotient of the free product`˚v We say that a group G decomposes as a graph of groups if there exists some graph of groups pG, Λq and spanning tree T of Λ such that Gπ 1 pG, Λ; T q.
Remark 2.25. Note that π 1 pG, Λ; T q can also be expressed as the quotient of π 1 pG, T ; T q˚F pEpΛq EpT qq by the normal subgroup generated by elements of the form e´1φé pgqeφè pgq´1, where e P EpΛq EpT q and g P G e ; this is immediate from the definition.
Remark 2.26. The group π 1 pG, Λ; T q does not depend on the choice of spanning tree T [Ser03, Proposition I.20]. We therefore often call π 1 pG, Λ; T q simply the fundamental group of pG, Λq and denote it π 1 pG, Λq.
We may also define a graph of groups decomposition as the fundamental group of a graph of spaces. Recall that a graph of pointed CW complexes is a system pX , Λq consisting of: vertex space) and pX e , x e q for each edge e P EpΛq (called an edge space); ‚ cellular maps pé : pX e , x e q Ñ pX opeq , x opeq q and pè : pX e , x e q Ñ pX tpeq , x tpeq q.
In particular, when each of the induced maps pé # : π 1 pX e , x e q Ñ π 1 pX opeq , x opeq q and pè # : π 1 pX e , x e q Ñ π 1 pX tpeq , x tpeq q is a monomorphism, we obtain a graph of groups pG, Λq where G v " π 1 pX v , x v q, G e " π 1 pX e , x e q, and φȇ " pȇ # .
Definition 2.27 (Total complex). The total complex associated to the graph of pointed CW complexes pX , Λq, denoted TotpX , Λq, is obtained by taking the disjoint union of the vertex spaces X v and the products X eˆr´1 , 1s of the edge spaces with intervals, then gluing these spaces using maps p e : X eˆt´1 , 1u Ñ X opeq \ X tpeq defined by p e px,˘1q " pȇ pxq.
Remark 2.29. We may apply this to a special cube complex X by cutting X along a collection of disjoint hyperplanes tH e u ePE ; that is, remove the open carrier of each H e to obtain a cube complex X 1 Ď X. Denote the connected components of X 1 by tX v u vPV and let X e " H eˆr´1 2 , 1 2 s be the closed carrier of H e . For each e P E, the two combinatorial hyperplanes associated to H e lie in some connected components X v and X w of X 1 , respectively (we may have v " w). Thus, to each e P E we may associate the pair pv, wq P VˆV . That is, V and E define the vertex set and edge set of some graph Λ, which is connected since X is connected. Furthermore, we may orient Λ and define maps pȇ by identifying H eˆt˘1 2 u Ď X e with the two combinatorial hyperplanes in X v and X w . This defines a graph of spaces pX , Λq such that X " TotpX , Λq.

Decomposing graph braid groups as graphs of groups
A graph braid group B n pΓq may be decomposed as graphs of groups by cutting Γ along a collection of edges that share a common vertex. In this section we show how to construct such a graph of groups decomposition. We begin by noting a result of Genevois, given below. Recall that we consider vertices of U C n pΓq to be subsets of V pΓq, as in Construction 2.13.
Lemma 3.1 ([Gen21b, Lemma 3.6]). Let E 1 and E 2 be two edges of U C n pΓq with endpoints S 1 , S 1 1 and S 2 , S 1 2 , respectively. Furthermore, let e i be the (closed) edge of Γ labelling E i for i " 1, 2. Then E 1 and E 2 are dual to the same hyperplane of U C n pΓq if and only if e 1 " e 2 ": e and S 1 X pΓ eq, S 2 X pΓ eq belong to the same connected component of U C n´1 pΓ eq. Convention 3.2. Note, when we write Γ Ω for some subgraph Ω Ď Γ, we mean the induced subgraph of Γ on the vertex set V pΓq V pΩq.
Remark 3.3 (Characterising combinatorial hyperplanes). Lemma 3.1 implies that each hyperplane H of U C n pΓq may be labelled by an edge e of Γ. Moreover, the combinatorial hyperplanes H˘associated to H are two subcomplexes of U C n pΓq isomorphic to the connected component of U C n´1 pΓ eq containing S X pΓ eq for some (hence any) S P pH˘q p0q . These two isomorphisms are obtained by fixing a particle at the two endpoints of e, respectively.
Remark 3.4 (Counting hyperplanes). Another immediate consequence of Lemma 3.1 is that the number of hyperplanes of U C n pΓq labelled by e P EpΓq is equal to the number of connected components of U C n´1 pΓ eq. If we let k e denote the number of connected components of Γ e, then Corollary 2.16 implies there are`n`k e´2 ke´1˘h yperplanes of U C n pΓq labelled by e. In other words, each hyperplane labelled by e corresponds to fixing one particle in e and partitioning the remaining n´1 particles among the connected components of Γ e.
We may combine Lemma 3.1 with Remark 2.29 to obtain graph of groups decompositions of a graph braid group where the vertex groups and edge groups are braid groups on subgraphs. The following theorem may be viewed as a strengthening of [Gen21b, Proposition 4.6].
Theorem 3.5. Let Γ be a finite connected graph (not necessarily satisfying the hypotheses of Theorem 2.17), let n ě 2, let m ě 1, and let e 1 , . . . , e m be distinct edges of Γ sharing a common vertex v. For each i P t1, . . . , mu, let H i be the collection of hyperplanes of U C n pΓq that are dual to edges of U C n pΓq labelled by e i , and let H " H 1 Y¨¨¨Y H m . The reduced graph braid group RB n pΓq decomposes as a graph of groups pG, Λq, where: ‚ V pΛq is the collection of connected components of U C n pΓ pe 1 Y¨¨¨Ye m qq; ‚ EpΛq " H, where H P EpΛq connects K, L P V pΛq (not necessarily distinct) if H has a combinatorial hyperplane in K and another in L; ‚ for each K P V pΛq, we have G K " RB n pΓ pe 1 Y¨¨¨Ye m q, S K q for some (equivalently any) basepoint S K P K; ‚ for each H i P H i Ď EpΛq, we have G H i " RB n´1 pΓ e i , S H i X pΓ e i qq, for some (equivalently any) vertex S H i of one of the combinatorial hyperplanes Hȋ ; ‚ for each edge H P EpΛq joining vertices K, L P V pΛq, the monomorphisms φH are induced by the inclusion maps of the combinatorial hyperplanes H˘into K and L.
Warning 3.6. One must be careful when computing vertex groups and edge groups in such a graph of groups decomposition, as removing edges from Γ may introduce new vertices of valence ‰ 2. This means that even if we assume Γ satisfies the hypotheses of Theorem 2.17, they may no longer hold when edges are removed. Thus, the reduced graph braid groups arising as vertex groups and edge groups may not be isomorphic to the corresponding graph braid groups even when RB n pΓq -B n pΓq.
Proof of Theorem 3.5. By Lemma 3.1 and Construction 2.13, two hyperplanes of U C n pΓq cross only if they are dual to edges of U C n pΓq labelled by disjoint edges of Γ. Since the edges e 1 , . . . , e m share a common vertex, the hyperplanes in H are therefore pairwise disjoint. By Remark 2.29, we may cut along H to obtain a graph of spaces decomposition pX , Λq of U C n pΓq (recall that U C n pΓq is special by Corollary 2.20). Moreover, by cutting along H, we are removing precisely the edges of U C n pΓq that are labelled by e i for some i. Thus, the cube complex we are left with is U C n pΓ pe 1 Y¨¨¨Ye m qq. The vertex spaces of pX , Λq are therefore the connected components of U C n pΓ pe 1 Y¨¨¨Ye m qq; that is, we may define V pΛq to be the collection of connected components of U C n pΓ pe 1 Y¨¨¨Ye m qq and take X K " K for each K P V pΛq. In particular, the vertex groups are G K " π 1 pX K q " π 1 pKq -π 1 pU C n pΓ pe 1 Y¨¨¨Ye m qq, S K q " RB n pΓ pe 1 Y¨¨¨Ye m q, S K q, where S K is any point in K.
Remark 2.29 further tells us that Λ has an edge between K, L P V pΛq (not necessarily distinct) for every hyperplane H P H that has a combinatorial hyperplane in K and another combinatorial hyperplane in L, and the associated edge space is Hˆr´1 2 , 1 2 s. Thus, we may define EpΛq " H and take X H " Hˆr´1 2 , 1 2 s for each H P EpΛq. Furthermore, Lemma 3.1 tells us that the combinatorial hyperplanes Hȋ associated to H i P H i are each isomorphic to the connected component of U C n´1 pΓ e i q containing S H i X pΓ e i q for some (hence any) S H i P pH˘q p0q . That is, the edge groups are G H i " π 1 pX H i q -π 1 pHȋ q -π 1 pU C n´1 pΓ e i q, S H i XpΓ e i qq " RB n´1 pΓ e i , S H i XpΓ e i qq.
Finally, Remark 2.29 tells us if H P EpΛq connects K, L P V pΛq, the cellular maps pH are obtained by identifying Hˆt˘1 2 u Ď X H with the two combinatorial hyperplanes H˘lying in K and L via the inclusion maps. Combinatorial hyperplanes of non-positively curved cube complexes are locally convex by Proposition 2.6, so H˘are locally convex in U C n pΓq and hence also in K and L. Thus, the maps pH are π 1 -injective by [HW08, Lemma 2.11]. In particular, each of the induced maps pH # is a monomorphism; denote these monomorphisms by φH.
In order to use this theorem to compute graph braid groups, we need to be able to determine when two vertices are connected by an edge, and also count the number of edges between each pair of vertices. We first fix some notation. Recall that the edges e i in Theorem 3.5 share a common vertex v; let v i denote the other vertex of e i . Let Γ v and Γ i denote the connected components of Γ pe 1 Y¨¨¨Ye m q containing v and v i , respectively.
Proposition 3.7 (Determining adjacency). Let pG, Λq be the graph of groups decomposition of RB n pΓq described in Theorem 3.5. Two vertices K, L P V pΛq are connected by an edge in H i Ď EpΛq if and only if the corresponding partitions given by Corollary 2.16 differ by moving a single particle from Γ v to Γ i (not necessarily distinct) along e i .
Proof. Let pG, Λq be a graph of groups decomposition of RB n pΓq arising from Theorem 3.5, and let K, L P V pΛq; that is, K and L are connected components of U C n pΓ pe 1 Y¨¨¨Ye m qq. Recall that by Corollary 2.16, connected components of U C n pΓ pe 1 Y¨¨¨Ye m qq correspond to partitions of the n particles among the connected components of Γ pe 1 Y¨¨¨Ye m q. Furthermore, K, L P V pΛq are connected by an edge H i P H i Ď EpΛq if and only if H i has a combinatorial hyperplane in K and another in L. By Remark 3.3, the partitions corresponding to K and L differ by moving a single particle from Γ v to Γ i along e i .
For each K, L P V pΛq connected by some edge in H i , let n K v , n K i and n L v , n L i denote the number of particles in Γ v , Γ i in the partitions corresponding to K and L, respectively. Let n v " maxtn K v , n L v u´1 and n i " maxtn K v , n L v u´1. Let k v , k i , and k v,i denote the number of connected components of Γ v v, Γ i v i , and Γ v pv Y v i q, respectively. We have the following result.
Proposition 3.8 (Counting edges). Let pG, Λq be the graph of groups decomposition of RB n pΓq described in Theorem 3.5, and suppose K, L P V pΛq are connected by some edge in H i . Let l i denote the number of edges in H i Ď EpΛq connecting K and L.
‚ If Γ v " Γ i , then l i is equal to the number of ways of partitioning the n v particles among the k v,i components of Γ v pv Y v i q. If each component of Γ v pv Y v i q has at least n v vertices, then l i "`n v`kv,i´1 Furthermore, the total number of edges of Λ connecting K and L is equal to Proof. Let pG, Λq be a graph of groups decomposition of RB n pΓq arising from Theorem 3.5, and suppose K, L P V pΛq are connected by an edge in H i . By Proposition 3.7, the partitions corresponding to K and L given by Corollary 2.16 differ by moving a single particle from Γ v to Γ i along e i . Applying Remark 3.4, the number of edges in H i connecting K and L is then computed by taking the partition corresponding to K, fixing this particle in e i , and then further partitioning the remaining particles in Γ v Y Γ i among the connected components of It then remains to sum over all j such that K and L are connected by an edge in H j . Note that if K and L are connected by an edge in both H i and H j , then by Proposition 3.7 the corresponding partitions differ both by moving a single particle from Γ v to Γ i along e i and by moving a single particle from Γ v to Γ j along e j . The only way this can happen is if Γ j " Γ i . Thus, the result follows.
3.1. Examples. We devote this section to the computation of specific graph braid groups by application of Theorem 3.5. Along the way, we answer a question of Genevois for all but one graph [Gen21b, Question 5.6].
Example 3.9. Consider the graphs ∆ and ∆ 1 shown in Figure 4 together with their corresponding cube complexes U C 2 p∆q and U C 2 p∆ 1 q, constructed by Algorithm 2.14. Note that ∆ is obtained from ∆ 1 by removing the interior of the edge e :" tv 1 , v 9 u.
We may therefore apply Theorem 3.5 to obtain a decomposition of B 2 p∆q as a graph of groups pG, Λq, where V pΛq is the collection of connected components of U C 2 p∆ eq " U C 2 p∆ 1 q. Since ∆ 1 is connected, it follows from Theorem 2.15 that U C 2 p∆ 1 q is also connected, hence Λ has a single vertex with associated vertex group RB 2 p∆ 1 q, which is isomorphic to B 2 p∆ 1 q since ∆ 1 satisfies the hypotheses of Theorem 2.17.
Furthermore, ∆ e is connected, so U C 1 p∆ eq is connected and hence by Lemma 3.1 there is only one hyperplane of U C 2 p∆q dual to an edge labelled by e. That is, Λ has a single edge with associated edge group RB 1 p∆ eq " π 1 p∆ eq -Z. We therefore see that B 2 p∆q is an HNN extension of B 2 p∆ 1 q; the corresponding graph of spaces decomposition of U C 2 p∆q is illustrated in Figure 5.
Observe that by collapsing certain hyperplane carriers onto combinatorial hyperplanes and collapsing certain edges to points, U C 2 p∆ 1 q deformation retracts onto a wedge sum of three circles C 1 , C 2 , C 3 and three tori T 1 , T 2 , T 3 , with successive tori glued along simple closed curves, denoted β and γ. This is illustrated in Figure 6. Figure 6. The cube complex U C 2 p∆ 1 q deformation retracts onto a wedge sum of three tori.
Proposition 3.10. Let n ě 2 and let R k be a k-pronged radial tree with k ě 3 (that is R k consists of k vertices of valence 1, each joined by an edge to a central vertex v of valence k). Then B n pR k q -F M , where M " M pn, kq " pk´2qˆn`k´2 k´1˙´ˆn`k´2 k´2˙`1 .
In particular, M ě 2 if and only if either n ě 3 or k ě 4.
Proof. First, subdivide each of the k edges of R k by adding n vertices of valence 2 along each edge, giving a graph R 1 k homeomorphic to R k that satisfies the hypotheses of Theorem 2.17. Let e 1 , . . . , e k denote the k edges of R 1 k sharing the central vertex v and apply Theorem 3.5 to e 1 , . . . , e k´2 to obtain a graph of groups decomposition pG, Λq of B n pR 1 k q. Note that R 1 k pe 1 Y¨¨¨Ye k´2 q and R 1 k pe 1 Y¨¨¨Y e k´2 q are both disjoint unions of segments, hence the vertex groups and edge groups of pG, Λq are all trivial. Thus, by Remark 2.25, B n pR 1 k q -B n pR k q is a free group F M where M " |EpΛq|´|EpT q| for some spanning tree T of Λ.
Note that R 1 k pe 1 Y¨¨¨Ye k´2 q has k´1 connected components: the component Γ v containing v, and the k´2 components Γ i containing the other vertex v i of e i . Thus, Λ has `n`k´2 k´2˘v ertices by Corollary 2.16, corresponding to the partitions of the n vertices among the k´1 connected components of R 1 k pe 1 Y¨¨¨Ye k´2 q. Denote the vertex with m particles in Γ v and m i particles in Γ i , i " 1, . . . , k´2, by K m,m 1 ,...,m k´2 . Then there are`N`k´3 k´3v ertices with m " n´N , corresponding to the partitions of the remaining N vertices among the k´2 components Γ i . By Proposition 3.7, K m,m 1 ,...,m k´2 and K l,l 1 ,...,l k´2 are connected by an edge of Λ if and only if |m´l| " 1, |m i´li | " 1 for some i, and m j " l j for all j ‰ i. To count the number of such pairs, fix m and note that K m,m 1 ,...,m k´2 is connected by an edge to the k´2 vertices of the form K m´1,m 1 ,...,m i`1 ,...,m k´2 . Thus, the number of pairs of vertices that are connected by an edge of Λ is recalling that ř r p"0`p`q q˘"`r`q`1 q`1˘f or any non-negative integers p, q, r. Moreover, since Γ v v has two connected components and Γ i v i has one connected component, Proposition 3.8 tells us the number of edges connecting the vertices K m,m 1 ,...,m k´2 and K m´1,m 1 ,...,m i`1 ,...,m k´2 is equal to`m´1`2´1 2´1˘" m. Thus, pk´2qˆn`k´2 k´1˙.
Proposition 3.11. Let n, k ě 3, r ď 2, and let R k,r be a k-pronged radial tree with k´r of its prongs subdivided by adding n vertices of valence 2 along them. Then RB n pR k,1 q -F M 1 and RB n pR k,2 q -F M 2 , where M 1 " pk´2qˆˆn`k´4 k´3˙`2ˆn`k´4 k´2˙˙´ˆn`k´2 k´2˙`1 M 2 " pk´2qˆˆn`k´3 k´2˙`ˆn`k´5 k´3˙˙´p k´3qˆn`k´6 k´2˙´ˆn`k´2 k´2˙`1 .
Proof. Following the proof of Proposition 3.10, let e 1 , . . . , e k denote the k edges of R k,r sharing the central vertex v, ordered so that the edges on subdivided prongs appear first, and apply Theorem 3.5 to e 1 , . . . , e k´2 to obtain a graph of groups decomposition pG, Λq of B n pR k,r q. That is, e 1 , . . . , e k´2 all lie on subdivided prongs. Note that R k,r pe 1 Y¨¨¨Ye k´2 q and R k,r pe 1 Y¨¨¨Y e k´2 q are both disjoint unions of segments, hence the vertex groups and edge groups of pG, Λq are all trivial. Thus, by Remark 2.25, RB n pR k,r q is a free group F Mr where M r " |EpΛq|´|EpT q| " |EpΛq|´|V pT q|`1 " |EpΛq|´|V pΛq|`1 for some spanning tree T of Λ.
The computations in the proof of Proposition 3.10 then apply, with the caveat that whenever we count partitions, we must bear in mind that the two prongs in Γ v may not have sufficiently many vertices to house all the particles. This affects the values of |V pΛq| and |EpΛq| as detailed below. As in the proof of Proposition 3.10, denote the vertex with m particles in Γ v and m i particles in Γ i , i " 1, . . . , k´2, by K m,m 1 ,...,m k´2 .
If r " 1, then Γ v still has at least n vertices, so Corollary 2.16 may be applied to show that Λ has`n`k´2 k´2˘v ertices, as in Proposition 3.10. Furthermore, by Proposition 3.8, the number of edges connecting the vertices K m,m 1 ,...,m k´2 and K m´1,m 1 ,...,m i`1 ,...,m k´2 is equal to the number of ways to partition the m´1 particles in Γ v among the two components of Γ v v. Since one component has only one vertex and the other has n vertices, this is equal to 1 if m " 1 and 2 if m ě 2. Thus, pk´2qˆˆn`k´4 k´3˙`2ˆn`k´4 k´2˙˙.
If r " 2, then Γ v has 3 vertices, so |V pΛq| is equal to the number of partitions of n particles into the k´1 components of R k,r pe 1 Y¨¨¨Ye k´2 q with either 0, 1, 2, or 3 particles in Γ v . That is, k´3"ˆn`k´2 k´2˙´ˆn`k´6 k´2˙.
For |EpΛq|, we are again restricted to the cases m " 0, 1, 2, 3. Moreover, when m " 3 there is only one way to partition the m´1 " 2 particles among the two components of Γ v v, since each component is a single vertex. Thus, k´3˙`ˆn`k´5 k´3˙" pk´2qˆˆn`k´3 k´2˙´ˆn`k´6 k´2˙`ˆn`k´5 k´3˙˙.
Proposition 3.12. Let Γ H be a segment of length 3 joining two vertices of valence three, as shown in Figure 7, so that Γ H is homeomorphic to the letter "H". Then B 4 pΓ H q -F 10˚Z 2 . Furthermore, let Γ 1 H be the graph obtained from Γ H by adding 4 vertices of valence 2 along one edge containing a vertex of valence 1 on each side of the segment of length 3. Then e Figure 7. The graph Γ H . We wish to split the corresponding graph braid group on 4 particles along the edge e.
Proof. Let e be the edge in the middle of the segment of length 3 and subdivide each of the four edges containing vertices of valence 1 by adding 4 vertices of valence 2 along each edge, giving a graph Γ 2 H homeomorphic to Γ H that satisfies the hypotheses of Theorem 2.17. Then Γ 2 H e is a disjoint union of two 3-pronged radial trees Γ 2 1 and Γ 2 2 with two prongs of each tree subdivided 4 times. That is, Γ 2 1 and Γ 2 2 are copies of R 3,1 . Furthermore, Γ 2 H e is a disjoint union of two segments Ω 2 1 and Ω 2 2 of length 4.
Applying Theorem 3.5 to B 4 pΓ 2 H q -B 4 pΓ H q, we obtain a graph of groups decomposition pG, Λq, where Λ has five vertices K 4,0 , K 3,1 , K 2,2 , K 1,3 , K 0,4 , corresponding to the five partitions of the 4 particles among the two subgraphs Γ 2 1 and Γ 2 2 ; that is, K i,j corresponds to the partition with i particles in Γ 2 1 and j particles in Γ 2 2 . Applying Proposition 3.11 and Lemma 2.18, we see that the vertex groups are given by Furthermore, Proposition 3.7 tells us that K i,j is adjacent to K k,l in Λ if and only if |i´k| " 1, and since Γ 2 H e has the same number of connected components as Γ 2 H e, by Proposition 3.8 there is at most one edge between each pair of vertices of Λ. Moreover, the edge groups are trivial since Ω 1 and Ω 2 are segments, which have trivial graph braid groups. Thus, we conclude that B 4 pΓ H q -F 3˚F2˚Z 2˚F 2˚F3 " F 10˚Z 2 . For RB 4 pΓ 1 H q, the computation is exactly the same, with the exception that Γ 1 H e is a disjoint union of two copies of R 3,2 rather than R 3,1 . Thus, Proposition 3.13. Let Γ A be a cycle containing two vertices of valence 3, as shown in Figure 8, so that Γ A is homeomorphic to the letter "A". Then B 4 pΓ A q -F 5˚Z 2 .
e Figure 8. The graph Γ A .
Proof. Let e be the edge indicated in Figure 8 and subdivide each of the two edges containing vertices of valence 1 by adding 4 vertices of valence 2 along each edge, giving a graph Γ 1 A homeomorphic to Γ A that satisfies the hypotheses of Theorem 2.17. Then Γ 1 A e " Γ 1 H . Applying Theorem 3.5 to B 4 pΓ 1 A q -B 4 pΓ A q, we obtain a graph of groups decomposition pG, Λq, where Λ has a single vertex since Γ 1 A e " Γ 1 H is connected, with vertex group RB 4 pΓ 1 H q. Furthermore, Γ 1 A e is a segment, therefore Λ has a single edge by Proposition 3.8, with trivial edge group. Thus, applying Proposition 3.12 and Remark 2.25, we have Proposition 3.14. Let Γ θ be the graph obtained by gluing two 6-cycles along a segment of length 3, as shown in Figure, so that Γ θ is homeomorphic to the letter "θ". Then B 4 pθq is an HNN extension of Z˚Z 2 .
Proof. Let e be the edge indicated in Figure 9 and note that Γ θ e " Γ A and Γ θ e is a 6-cycle. Applying Theorem 3.5 and Proposition 3.8 to B 4 pΓ θ q, we obtain a graph of groups decomposition pG, Λq, where Λ has a single vertex with vertex group RB 4 pΓ A q and a single edge with edge group Z. Thus, B 4 pΓ θ q is an HNN extension of RB 4 pΓ A q. Furthermore, we may apply Theorem 3.5 and Proposition 3.8 to RB 4 pΓ A q using the edge indicated in Figure 8, noting that Γ A e " Γ H and Γ A e is a segment. Thus, we obtain a graph of groups decomposition pG 1 , Λ 1 q, where Λ 1 has a single vertex with vertex group RB 4 pΓ H q and a single edge with trivial edge group. Thus, RB 4 pΓ A q -RB 4 pΓ H q˚Z.
Finally, apply Theorem 3.5 and Propositions 3.7 and 3.8 to RB 4 pΓ H q using the edge indicated in Figure 7, noting that Γ H e is a disjoint union of two copies of the 3-pronged radial tree R 3 and Γ H e is a disjoint union of two segments. Thus, we obtain a graph of groups decomposition pG 2 , Λ 2 q, where Λ 2 has five vertices K 4,0 , K 3,1 , K 2,2 , K 1,3 , K 0,4 corresponding to the five partitions of the 4 particles among the two copies of R 3 , with K i,j adjacent to K k,l in Λ 2 if and only if |i´k| " 1, in which case they are connected by a single edge.
Moreover, the vertex group associated to K i,j is RB i pR 3 qˆRB j pR 3 q by Lemma 2.18, and the edge groups are all trivial. Note that: U C 0 pR 3 q is a single point hence RB 0 pR 3 q is trivial; U C 1 pR 3 q " R 3 hence RB 1 pR 3 q is trivial; RB 2 pR 3 q -B 2 pR 3 q -Z by Proposition 3.10; and U C 4 pR 3 q is a single point since R 3 has only 4 vertices, so RB 4 pR 3 q is trivial. Furthermore, since R 3 has 4 vertices, we have U C 3 pR 3 q " U C 1 pR 3 q. This can be seen by considering configurations of particles on vertices of R 3 as colourings of the vertices of R 3 in black or white according to whether they are occupied by a particle. By inverting the colouring, we see that U C 3 pR 3 q " U C 1 pR 3 q. Thus, RB 3 pR 3 q is trivial. It therefore follows that RB 4 pΓ H q -Z 2 .
We therefore conclude that B 4 pΓ θ q is an HNN extension of RB 4 pΓ A q -RB 4 pΓ H q˚Z - Note that Propositions 3.12, 3.13, and 3.14 answer [Gen21b, Question 5.6], up to determining whether B 4 pΓ θ q is a non-trivial free product.
Theorem 3.15. Let Γ be a finite connected graph. The braid group B 4 pΓq is toral relatively hyperbolic only if it is either a free group or isomorphic to F 10˚Z 2 or F 5˚Z 2 or an HNN extension of Z˚Z 2 .
Proof. By [Gen21b, Theorem 4.22], B 4 pΓq is toral relatively hyperbolic if and only if Γ is either: a collection of cycles and segments glued along a single central vertex (called a flower graph); a graph Γ H homeomorphic to the letter "H"; a graph Γ A homeomorphic to the letter "A"; or a graph Γ θ homeomorphic to the letter "θ". Braid groups of flower graphs are free by [Gen21b,Corollary 4.7], while B 4 pΓ H q and B 4 pΓ A q are isomorphic to F 10˚Z 2 and F 5˚Z 2 respectively, by Propositions 3.12 and 3.13, and B 4 pΓ θ q is an HNN extension of Z˚Z 2 by Proposition 3.14.
Finally, we compute the braid group of the graph homeomorphic to the letter "Q", for any number of particles.
Proposition 3.16. Let Γ Q be a cycle containing one vertex of valence 3, as shown in Figure  10, so that Γ Q is homeomorphic to the letter "Q". Then B n pΓ Q q -F n .
e Figure 10. The graph Γ Q .
Proof. Let e be the edge indicated in Figure 10, and subdivide all other edges of Γ Q by adding n vertices of valence 2 along them, so that the hypotheses of Theorem 2.17 are satisfied. Note that Γ Q e is a segment, while Γ Q e is a disjoint union of two segments. Applying Theorem 3.5 and Proposition 3.8 to B n pΓ Q q, we obtain a graph of groups decomposition pG, Λq, where Λ has a single vertex with trivial vertex group and`n 1˘" n edges with trivial edge groups. Thus, Remark 2.25 tells us that B n pΓ Q q -F n .

4.1.
Decomposing graph braid groups as free products. In Section 3.1, we saw that it appears to be quite common for a graph braid group to decompose as a non-trivial free product. We now prove two general criteria for a graph braid group to decompose as a nontrivial free product, and apply these to answer a question of Genevois [Gen21b, Question 5.3].
We define the following classes of graphs (c.f. [Gen21b, Section 4.1]). See Figure 11 for examples.
‚ A flower graph is a graph obtained by gluing cycles and segments along a single central vertex. ‚ A sun graph is a graph obtained by gluing segments to vertices of a cycle. ‚ A pulsar graph is obtained by gluing cycles along a fixed non-trivial segment Ω and gluing further segments to the endpoints of Ω. ‚ The generalised theta graph Θ m is the pulsar graph obtained by gluing m cycles along a fixed segment. Figure 11. Left to right: a flower graph, a sun graph, a pulsar graph, and Θ 2 .
Lemma 4.1 (Free product criterion 1). Let n ě 2 and let Γ be a finite graph obtained by gluing a non-segment flower graph Φ to a connected non-segment graph Ω along a vertex v, where v is either the central vertex of Φ or a vertex of Φ of valence 1. Then B n pΓq -H˚Z for some non-trivial subgroup H of B n pΓq.
If v ‰ v c and Φ is not a 3-pronged radial tree, then we may remove the segment connecting v to v c from Φ and add it to Ω. The new Φ and Ω will still satisfy the hypotheses of the lemma, and Φ X Ω " v c . If Φ is a 3-pronged radial tree, then after subdividing edges sufficiently we may remove all but the last edge of the segment connecting v to v c from Φ and add it to Ω, so that Φ X Ω " v where v is a vertex of valence 2 in Γ adjacent to v c . Thus, henceforth we shall assume without loss of generality that Φ X Ω " v where either v " v c or Φ is a 3-pronged radial tree and v is a vertex of valence 2 in Γ adjacent to v c . Let e 1 , . . . , e m denote the edges of Ω that have v as an endpoint and apply Theorem 3.5 to obtain a graph of groups decomposition pG, Λq of B n pΓq. Note that Γ pe 1 Y¨¨¨Ye m q is a disjoint union of Φ and some subgraphs Ω 1 , . . . , Ω k of Ω. By sufficiently subdividing the edges of Γ, we may assume without loss of generality that the graphs Γ, Φ, Ω 1 , . . . , Ω k all satisfy the hypotheses of Theorem 2.17. The vertices of Λ correspond to the partitions of the n particles among Φ, Ω 1 , . . . , Ω k . Let K p,p 1 ,...,p k denote the partition with p particles in Φ and p i particles in Ω i , i " 1, . . . , k. Proposition 3.7 tells us that K :" K n,0,...,0 is adjacent to precisely the vertices K 1 i :" K n´1,0,...,1,...,0 for i " 1, . . . , k, where K 1 i is the partition with n´1 particles in Φ and one particle in Ω i . Furthermore, K 1 i is adjacent to precisely K and the vertices K 1,1 i,j for j ‰ i and K 2 i , where K 1,1 i,j is the partition with n´2 particles in Φ, one particle in Ω i and one particle in Ω j , and K 2 i is the partition with n´2 particles in Φ and two particles in Ω i .
First suppose Ω i contains a cycle for some i. We have G K -B n pΦq, which is a free group by [Gen21b,Corollary 4.7] and is non-trivial by Lemma 2.22 since n ě 2 and Φ is not a segment. We also have G K 1 i -B n´1 pΦqˆB 1 pΩ i q by Lemma 2.18, which is non-trivial for some i by Lemma 2.22 since some Ω i contains a cycle. Furthermore, for each i the edges connecting K to K 1 i all have edge groups isomorphic to RB n´1 pΦ vq, which is trivial since Φ v is a disjoint union of segments. Thus, by Remark 2.25 B n pΓq decomposes as a free product of the form B n pΓq -H˚B n pΦq -H˚F -H 1˚Z for some non-trivial free group F and non-trivial group H. Now suppose that Ω i does not contain a cycle for any i.
If Ω i are all segments, then Ω is a non-segment flower graph. Furthermore, v is either the central vertex of Ω (in which case v has valence ě 3 in Γ and so we must have v " v c ) or Ω is a 3-pronged radial tree and v is a vertex of Ω of valence 1. In the former case, Γ is a flower graph containing a vertex of valence ě 4, thus B n pΓq is a free group by [Gen21b,Corollary 4.7], and by Lemma 2.21 and Proposition 3.10 its rank is at least 2. That is, B n pΓq -H˚Z for some non-trivial subgroup H, as required. In the latter case, by subdividing edges of Ω we may assume Ω v is also a 3-pronged radial tree. Thus, we may assume some Ω i contains a vertex of valence ě 3.
Note that for each i, the edges connecting K 1 i to K 1,1 i,j and the edges connecting K 1 i to K 2 i all have edge groups isomorphic to RB n´2 pΦ vqˆRB 1 pΩ i vq, which is trivial by Lemma 2.22 since Φ v is a disjoint union of segments and Ω i v does not contain a cycle. Furthermore, we have G K 2 i -B n´2 pΦqˆB 2 pΩ i q, which is non-trivial for some i by Lemma 2.22 since some Ω i contains a vertex of valence ě 3. Thus, by Remark 2.25 we again have B n pΓq -H˚Z, for some non-trivial group H.
Lemma 4.2 (Free product criterion 2). Let n ě 2 and let Γ be a finite graph satisfying the hypotheses of Theorem 2.17. Suppose Γ contains an edge e such that Γ e is connected but Γ e is disconnected, and one of the connected components of Γ e is a segment of length at least n´2. Then B n pΓq -H˚Z for some non-trivial subgroup H of B n pΓq.
Proof. First suppose Γ e is a segment. Then Γ must consist of a single cycle, containing e, with a segment glued to either one or both endpoints of e (there must be at least one such segment since Γ e is disconnected). That is, Γ is homeomorphic to either Γ Q or Γ A -the graphs homeomorphic to the letters "Q" and "A", respectively.
If Γ is homeomorphic to Γ Q , then Proposition 3.16 tells us B n pΓq -F n . Since n ě 2, this implies B n pΓq -H˚Z for some non-trivial subgroup H. If Γ is homeomorphic to Γ A , then we may subdivide the edge e to obtain a graph Γ 1 homeomorphic to Γ and containing no edge e 1 such that Γ 1 e 1 is a segment. Furthermore, Γ 1 still contains an edge e 2 such that Γ 1 e 2 is connected but Γ 1 e 2 is disconnected, and one of the connected components is a segment. Thus, we may assume without loss of generality that we are in the case where Γ e is not a segment.
Suppose Γ e is not a segment. Applying Theorem 3.5, we obtain a graph of groups decomposition pG, Λq of B n pΓq where Λ has a single vertex with vertex group RB n pΓ eq, which is non-trivial by Lemma 2.22 since Γ e is not a segment. Furthermore, Γ e is disconnected with a segment Ω as one of its connected components. Let S P U C n pΓ eq p0q be a configuration with one particle at an endpoint of e and all remaining n´1 particles in the segment Ω; this configuration exists since Ω has length at least n´2. Then Λ has an edge with edge group RB n´1 pΓ e, S X pΓ eqq, which is isomorphic to RB n´1 pΩq " 1 by Lemma 2.18. Thus, Remark 2.25 tells us B n pΓq decomposes as a free product B n pΓq -H˚Z, where H is non-trivial since the vertex group RB n pΓ eq is non-trivial.
Note that in particular, Lemma 4.2 implies that all braid groups on sun graphs decompose as non-trivial free products, with the exception of the cycle graph, which has graph braid group isomorphic to Z. Lemma 4.2 also implies that all braid groups on pulsar graphs decompose as non-trivial free products, with the possible exception of generalised theta graphs Θ m for m ě 2. Thus, we have partially answered [Gen21b, Question 5.3].
Theorem 4.3. Let Γ be a finite connected graph that is not homeomorphic to Θ m for any m ě 0. The braid group B 3 pΓq is hyperbolic only if B 3 pΓq -H˚Z for some group H.
Proof. By [Gen21b, Theorem 4.1], B 3 pΓq is hyperbolic if and only if Γ is a tree, a flower graph, a sun graph, or a pulsar graph. If Γ is a tree, then either Γ is a radial tree, in which case B 3 pΓq is free of rank M ě 3 by Proposition 3.10, or Γ has at least two vertices of valence ě 3, in which case B 3 pΓq -H˚Z by Lemma 4.1. In fact, B 3 pΓq is known to be free by [FS05,Theorems 2.5,4.3]. If Γ is a flower graph, then B 3 pΓq is free by [Gen21b,Corollary 4.7]. Suppose Γ is a sun graph or a pulsar graph that contains at least one cycle and is not homeomorphic to Θ m for any m. Subdivide edges of Γ so that it satisfies the hypotheses of Theorem 2.17. Then Γ has an edge e that is contained in a cycle and incident to a vertex of degree at least 3 with a segment of length at least 2 glued to it. Thus, Γ e is connected but Γ e is disconnected, and one of the connected components of Γ e is a segment of length at least 1. Thus, B 3 pΓq -H˚Z for some group H by Lemma 4.2. If Γ is a pulsar graph that contains no cycles and is not homeomorphic to Θ 0 , then Γ consists of a segment with further segments glued to the endpoints. Thus, we may apply Lemma 4.1 to conclude that B 3 pΓq -H˚Z for some subgroup H.

4.2.
Pathological relative hyperbolicity in graph braid groups. We can apply Theorem 3.5 to subgraphs of the graph ∆ 1 given in Figure 4 to provide a negative answer to a question of Genevois regarding relative hyperbolicity of graph braid groups [Gen21b, followup to Question 5.7].
Theorem 4.4. Let ∆ 1 be the graph given in Figure 4. The graph braid group B 2 p∆ 1 q is hyperbolic relative to a thick, proper subgroup P that is not contained in any graph braid group of the form B k pΛq for k ď 2 and Λ Ĺ ∆ 1 .
The group B 2 p∆ 1 q is therefore hyperbolic relative to P " π 1 pT 1 q˚x by π 1 pT 2 q˚x cy π 1 pT 3 q.
Note that P is the group constructed by Croke  Suppose P is contained in a graph braid group of the form B k pΛq for k ď 2 and Λ Ĺ ∆ 1 . Since P does not split as a direct product, we may assume that Λ is connected by Lemma 2.18. Moreover, since P contains a Z 2 subgroup, we may assume by Theorem 2.23 that k " 2 and Λ contains two disjoint cycles. We may therefore deduce that Λ must be isomorphic to one of the six graphs Λ i , i " 1, . . . , 6, given in Figure 12.
Since P -A Π does not split as a non-trivial free product and does not embed as a subgroup of Z 2 , we conclude that P cannot be contained in B 2 pΛ 3 q or B 2 pΛ 6 q by the Kurosh subgroup theorem [Kur34,Untergruppensatz]. Since Λ 1 and Λ 2 are proper subgraphs of Λ 3 , and Λ 4 and Λ 5 are proper subgraphs of Λ 6 , we see by Lemma 2.21 that P cannot be contained in B 2 pΛ i q for any 1 ď i ď 6. Thus, P cannot be contained in any graph braid group of the form B k pΛq for k ď 2 and Λ Ĺ ∆ 1 .
This provides a negative answer to a question of Genevois [Gen21b, follow-up to Question 5.7], showing that while peripheral subgroups arise as braid groups of subgraphs in the case of toral relative hyperbolicity, this is not true in general.
Remark 4.5 (Characterising relative hyperbolicity via hierarchical hyperbolicity). Graph braid groups are hierarchically hyperbolic; since they are fundamental groups of special cube complexes, one may apply [BHS17, Proposition B, Remark 13.2] to show this. Indeed, a natural explicit hierarchically hyperbolic structure is given in [Ber21, Section 5.1.2], described in terms of braid groups on subgraphs.
Furthermore, Russell shows that hierarchically hyperbolic groups (HHGs) are relatively hyperbolic if and only if they admit an HHG structure S with isolated orthogonality [Rus20, Corollary 1.2], a simple criterion restricting where orthogonality may occur in the HHG structure. Roughly, this means there exist I 1 , . . . , I n P S that correspond to the peripheral subgroups.
One may therefore hope that relative hyperbolicity of graph braid groups could be characterised via this approach. However, the above example is problematic here, too; since the peripheral subgroup P does not arise as a braid group on a subgraph of ∆ 1 , it does not appear in the natural HHG structure described in [Ber21]. Thus, Russell's isolated orthogonality criterion is not able to detect relative hyperbolicity of B 2 p∆ 1 q through this HHG structure. 4.3. Distinguishing right-angled Artin groups and graph braid groups. Note that Lemmas 4.1 and 4.2 preclude certain right-angled Artin groups and graph braid groups from being isomorphic.
Theorem 4.6. Let n ě 2, let Π be a connected graph and let Γ be a finite graph such that one of the following holds: ‚ Γ is obtained by gluing a connected non-trivial graph Ω to a non-segment flower graph Φ along the central vertex of Φ; ‚ Γ satisfies the hypotheses of Theorem 2.17 and contains an edge e such that Γ e is connected but Γ e is disconnected, and one of the connected components of Γ e is a segment of length at least n´2.
Then A Π is not isomorphic to B n pΓq.
Proof. Since Π is connected, the right-angled Artin group A Π does not decompose as a nontrivial free product. However, by Lemmas 4.1 and 4.2, we know that B n pΓq must decompose as a non-trivial free product.
In fact, by combining Lemmas 4.1 and 4.2 with a result of Kim-Ko-Park [KKP12, Theorem B], we obtain stronger results for graph braid groups on large numbers of particles.
Theorem 4.7. Let Π and Γ be finite connected graphs with at least two vertices and let n ě 5. Then the right-angled Artin group A Π is not isomorphic to B n pΓq.
Proof. By [KKP12,Theorem B], if Γ contains the graph Γ A homeomorphic to the letter "A", then B n pΓq is not isomorphic to a right-angled Artin group. Thus, by sufficiently subdividing edges of Γ, we may assume that all cycles in Γ contain at most one vertex of valence ě 3.
If Γ contains a cycle C with precisely one vertex v of valence ě 3, then we can express Γ as the graph C glued to the graph Ω " Γ pC vq along the vertex v. If Ω is a segment, then Γ is homeomorphic to the letter "Q", thus Proposition 3.16 tells us B n pΓq -F n . Otherwise, we can apply Lemma 4.1 to show that B n pΓq -H˚Z for some non-trivial subgroup H of B n pΓq. However, A Π cannot be expressed as a non-trivial free product, since Π is connected. Thus, in both cases it follows that A Π is not isomorphic to B n pΓq. We may therefore assume that no cycles in Γ contain a vertex of valence ě 3.
If Γ contains a cycle C with no vertices of valence ě 3, then Γ is either disconnected or Γ " C. The former case is excluded by hypothesis, while the latter implies B n pΓq -Z. In particular, since Π has at least two vertices, this implies B n pΓq is not isomorphic to A Π .
If Γ contains no cycles, then Γ is a tree. If Γ is a segment, then B n pΓq is trivial. If Γ is homeomorphic to a k-pronged radial tree R k , then Proposition 3.10 tells us B n pΓq -F M for some M ě 10. Otherwise, Γ contains at least two vertices of valence ě 3. Thus, Γ can be expressed as Γ " Φ Y Ω, where Φ is a segment, Ω is not a segment, and Φ X Ω is a vertex of valence ě 2. Thus, we may apply Lemma 4.1 to conclude that B n pΓq -H˚Z for some non-trivial subgroup H, hence B n pΓq is not isomorphic to A Π .

Open questions
Theorem 4.3 depends on the assumption that Γ is not homeomorphic to a generalised theta graph Θ m . Furthermore, Proposition 3.14 tells us that B 4 pΓ θ q is an HNN extension of Z˚Z 2 , but it is not clear whether it is a non-trivial free product; knowing this would benefit Theorem 3.15. Computing braid groups of generalised theta graphs would therefore be very helpful to improve these two theorems.
Question 5.1. What is B n pΘ m q for n ě 3 and m ě 2? Are they non-trivial free products?
Note, it is easy to show B 2 pΘ m q is a free group of rank at least 3 using Theorem 3.5, however the case of n ě 3 is trickier. One potential way of approaching this problem would be to apply Theorem 3.5, letting v be one of the vertices of Θ m of valence m`1 and decomposing B n pΘ m q as a graph of groups pG, Λq using m of the edges incident to v. Denoting these edges by e 1 , . . . , e m , the graphs Θ m pe 1 Y¨¨¨Ye m q and Θ m pe 1 Y¨¨¨Ye m q are both homeomorphic to the pm`1q-pronged radial tree R m`1 , as illustrated in Figure 14. Thus, by Propositions 3.8 and 3.10, Λ has a single vertex whose vertex group is a free group of rank M pn, m`1q, and m edges whose edge groups are free groups of rank M pn´1, m`1q, where M is defined in Proposition 3.10. It remains to understand the monomorphisms mapping the edge groups into the vertex groups. Question 5.2. Are there any graph braid groups that decompose as non-trivial free products but do not satisfy the hypotheses of Lemma 4.1 or Lemma 4.2?
Theorems 4.6 and 4.7 also raise a natural question: can right-angled Artin groups with connected defining graphs be isomorphic to graph braid groups B n pΓq for n ď 4? Question 5.3. Does there exist some n ě 2 and a finite connected graph Γ such that B n pΓq is isomorphic to a non-cyclic right-angled Artin group with connected defining graph?
Finally, Theorem 4.4 shows that the peripheral subgroups of a relatively hyperbolic graph braid group B n pΓq may not be contained in any braid group B k pΛq for Λ Ĺ Γ. Thus, while the groups B k pΛq form a natural collection of subgroups of B n pΓq by Lemma 2.21, they are insufficient to capture the relatively hyperbolic structure. The question of how to classify relative hyperbolicity in graph braid groups therefore remains.
Question 5.4. When is a graph braid group relatively hyperbolic?
One potential approach would be to construct a hierarchically hyperbolic structure on B n pΓq with greater granularity than the one produced in [Ber21], and then apply Russell's isolated orthogonality criterion as described in Remark 4.5. In particular, one would have to construct a factor system on the cube complex U C n pΓq that contains enough subcomplexes to guarantee that any peripheral subgroup of B n pΓq will always appear as the fundamental group of such a subcomplex.