Commensurating HNN-extensions: hierarchical hyperbolicity and biautomaticity

We construct a CAT(0) hierarchically hyperbolic group (HHG) acting geometrically on the product of a hyperbolic plane and a locally-finite tree which is not biautomatic. This gives the first example of an HHG which is not biautomatic, the first example of a non-biautomatic CAT(0) group of flat-rank 2, and the first example of an HHG which is injective but not Helly. Our proofs heavily utilise the space of geodesic currents for a hyperbolic surface.


Introduction
Let H be a locally compact group with Haar measure µ.A discrete subgroup Γ ă H is a lattice if µpH{Γq is finite.We say Γ is uniform is H{Γ is compact.Roughly speaking, we say a lattice Γ ă H 1 ˆH2 is irreducible if the projection of Γ Ñ H i is non-discrete and if Γ does not split as a direct product of two infinite groups (see Section 2 for details).Throughout we will denote the n-regular tree by T n and its automorphism group by T n .
Automatic and biautomatic groups were developed in the 1980s; with a detailed account given in the book [Eps+92] by Epstein, Cannon, Holt, Levy, Paterson, and Thurston.In the 1990s Alonso and Bridson introduced the class of semihyperbolic groups [AB95] which contains all CATp0q and biautomatic groups.In recent work of Leary and Minasyan [LM21], the authors construct irreducible uniform lattices in IsompE 2n q ˆT2m (m ě 2, n ě 1), giving the first examples of CATp0q groups which are not biautomatic.These groups were classified up to isomorphism by the second author [Val21a] and studied in the context of fibring by the first author [Hug22a].It follows from [Hug21b] and [Val21b] that all known examples of CATp0q but not biautomatic groups are either constructed from or contain non-biautomatic Leary-Minasyan groups as subgroups.
In the 2010s, the coarse geometric class of hierarchically hyperbolic groups (HHGs) and spaces (HHSs) were introduced by Behrstock, Hagen and Sisto in [BHS17a] with the motivation coming from isolating the main geometric features common to mapping class groups and compact special groups.
Very roughly these are spaces admitting a coordinate system and hierarchy consisting of and parameterised by hyperbolic spaces, and groups of isometries acting geometrically whilst preserving the hierarchy and coordinate structure.The theory has received a lot of attention; being studied and developed by numerous authors [BHS17b; DHS17; DHS17; Spr18a; Spr18b; AB19; ANS19; BR19; DMS20; RS20; PS22].
As previously mentioned some of the main motivation for, and examples of, HHGs come from CATp0q cubical groups [BHS17b;HS20] which are known to be biautomatic by the work of Niblo and Reeves [NR98].A 2021 result of Haettel, Hoda and Petyt shows that HHGs are semihyperbolic [HHP20], as a corollary this gave a new proof that mapping class groups are semihyperbolic (see also [DMS20] and [Ham09]).One may hope that proving HHGs are biautomatic would give another proof that mapping class groups are biautomatic.Thus, a natural question is whether every HHG is biautomatic?It appears to be open whether any non-biautomatic Leary-Minasyan groups are HHGs-although experts expect them not to be.In this paper we construct the first example of an HHG which is not biautomatic.
Theorem A. There exists a non-residually finite torsion-free uniform irreducible lattice Γ ă PSL 2 pRq ˆT24 such that Γ is a hierarchically hyperbolic group but is not biautomatic.
The group we construct is a "hyperbolic" analogue of the groups introduced by Leary-Minasyan in [LM21].Indeed, Γ is an HNN-extension of an arithmetic surface where the stable letter commensurates the surface whilst acting as an infinite order elliptic isometry of the hyperbolic plane RH 2 .That the action is by isometries allows us to deduce that Γ is a CATp0q lattice acting freely cocompactly on the product RH 2 ˆT24 , where T 24 is the Bass-Serre tree.Note that we adopt the lattice viewpoint so we may use results of [Hug21b].From here we apply [Hug22b, Corollary 3.3] to deduce Γ is an HHG.
Our strategy to show that Γ is not biautomatic is very different to Leary-Minasyan's work (for example Γ is neither constructed from nor contains a Leary-Minasyan group).Instead of studying the boundary of a biautomatic structure, we develop a new method to show the failure of biautomaticity.In particular, we use deep work of Martínez-Granado and Thurston on extending functions to the space of geodesic currents of a hyperbolic surface [Mar20;MT21].
The question of whether every automatic group is biautomatic first appeared in [Eps+92, Question 2.5.6] and [GS91,Remark 6.19].We do not know if the group Γ is automatic.In spite of this we can still deduce an amusing consequence.
Corollary B. At least one of the following statements is false: (1) Every HHG is automatic.
Note that the analogous statement with "CATp0q group" instead of "HHG" follows from the work of Leary and Minasyan [LM21].However, since Γ is CATp0q it can be deduced here too.
Recall that the flat-rank of a CATp0q group Γ (acting on X), denoted flat-rankpΓq, is the maximal rank of an isometrically embedded Euclidean space in X.
In [FHT11,Question 43] it was asked if every group acting geometrically on a piece-wise Euclidean CATp0q 2-complex group is biautomatic.(See [FHT11] and [MOP21] for recent progress.)One may hope to relax the hypothesis "2-dimensional piece-wise Euclidean CATp0q" to "flat-rank 2 CATp0q".Indeed, all previous examples of CATp0q but not biautomatic groups have had flat-rank at least 3.The next corollary, which follows from the Flat Torus Theorem, shows that one cannot.
Corollary C.There exists a CATp0q group Γ with flat-rankpΓq " 2, that is not biautomatic.
In [HP20] the authors introduce a property regarding commensurators of abelian subgroups, Condition (C), and show that its failure is closely related to Leary-Minasyan groups [HP20,Proposition 8.4].A natural question would be to ask whether the failure of Condition (C) for a CATp0q group is equivalent to the failure of biautomaticity.However, by [HP20, Theorem 1.3], the group Γ has Condition (C) and fails to be biautomatic.
In [BHS19,Theorem 7.3] it is shown that HHGs are coarse median spaces as introduced by Bowditch [Bow13].We say a group is a coarse median group if it acts geometrically on a coarse median space and the coarse median operator is equivariant up to bounded error.In [Pet22, Remark 3.14] it is shown that HHGs are coarse median groups.We remark that Γ appears to be the first example of a coarse median group of type F which is not biautomatic.
The group Γ also appears as an example highlighting the difference between discrete and non-discrete versions of "injective" metric spaces.We say that a geodesic metric space (respectively a graph) X is injective (respectively Helly) if the collection of all metric balls in X satisfies the Helly property.Injective metric spaces and Helly graphs, as well as groups acting on them geometrically-injective groups and Helly groups, respectivelyhave been extensively studied [Isb64; Dre84; Lan13; DL16; BC08; Cha+20; HO21].The following result gives a negative answer to the question in [Hae21, Page 4].
Corollary D. There exists a group Γ which is injective but not Helly and not biautomatic.
Proof.It follows from Theorem A that Γ is not biautomatic.Moreover, Helly groups are biautomatic [Cha+20, Theorem 1.5(1)], and so Γ is not Helly.On the other hand, for every metric space X there exists a "smallest" injective metric space EpXq, called the injective hull of X, into which X embeds isometrically, so that a group action on X extends to an action on EpXq [Isb64].It is known that EpRH 2 q is proper and finite Hausdorff distance away from the image of RH 2 ãÑ EpRH 2 q [Hae21, Proposition 4.6]; it also follows from the definitions that (real) trees are injective and that the 8 product X ˆ8 Y of injective spaces X and Y is injective.Therefore, the geometric action of Γ on RH 2 ˆ8 T 24 extends to a geometric action on the proper injective metric space EpRH 2 q ˆ8 T 24 , and so Γ is injective, as required.
On the other hand, one may replace the Helly property with a coarse Helly property to study the classes of coarsely injective and coarsely Helly graphs and groups.Coarsely injective and coarsely Helly groups have been studied in [HHP20; Cha+20; OV20]; in particular, it has been shown that all HHGs are coarsely injective [  .It would be extremely interesting to adapt the methods here to apply to a uniform S-arithmetic lattice in PSL 2 pRq ˆPSL 2 pQ p q.The main issue is showing that vertex stabilisers in the action on the Bruhat-Tits tree T p`1 are quasi-convex with respect to any biautomatic structure on the lattice.Note that since such a lattice is residually finite, so if this strategy can be implemented successfully, one would also get a negative answer to [LM21,Question 12.4].
We end with a broad conjecture which would vastly generalise our work here.The reader is directed to Section 2 for definitions.
Conjecture 1.3.Let H be a semi-simple real Lie group with trivial centre and no compact factors.Let T be the automorphism group of a locally-finite unimodular leafless tree.Suppose T is non-discrete.If Γ is an irreducible non-residually finite uniform pH ˆT q-lattice, then Γ is not biautomatic.
1.1.Outline of the paper.In Section 2 we revise the necessary background on lattices, biautomatic structures, geodesic currents on a hyperbolic surface, and the intersection form.The remainder of the article is then dedicated to proving Theorem A.
The strategy of the proof of Theorem A is as follows.We first assume that Γ has a biautomatic structure pB, Mq and consider a biautomatic structure pA, Lq induced by pB, Mq on a quasi-convex subgroup G; here G is a vertex stabiliser in the action of Γ on T .The group G acts freely cocompactly on a copy of RH 2 and so can be identified with a subgroup of PSL 2 pRq, giving rise to a Riemann surface Σ " GzPSL 2 pRq.The next step is to show that stable word length function τ L : G Ñ R with respect to pA, Lq takes only rational values and extends over the space of geodesic currents of Σ.Now, the translation length function of G also extends to the space of geodesic currents of Σ.Moreover, using the density of the projection of Γ to PSL 2 pRq we show that both functions agree.Now, the translation length function takes values which are not rational multiples of each other.This is a contradiction, and so Γ cannot be biautomatic.
In Section 3 we study stable word length τ L on a biautomatic structure pA, Lq as a function from G Ñ R. The key results, Proposition 3.1 and Lemma 3.5, imply that for a hyperbolic group G the function takes rational values.
In Section 4 we show that the function τ L , viewed as a function on the homotopy classes of closed curves on Σ, satisfies a technical property known as "quasi-smoothing" (see Proposition 4.2).This allows us to extend τ L continuously to the space of geodesic currents of Σ.
In Section 5 we complete our study of functions on geodesic currents.The key result, Proposition 5.6, is that if t is an elliptic isometry of RH 2 commensurating G such that xG, ty is dense in PSL 2 pRq, and if a continuous function F on the space of geodesic currents of Σ is in a sense "t-invariant", then F pγq is a constant multiple of the length of the geodesic representative of γ, where γ is a closed curve on Σ.In the remaining sections, we construct a lattice Γ ă PSL 2 pRq ˆT that will allow us to apply this result for F " τ L .
In Section 6 we study properties of irreducible uniform lattices in PSL 2 pRqT for sufficiently general trees.In particular, for a non-residually finite lattice we prove that projection to PSL 2 pRq is dense (Lemma 6.1) and that a vertex stabiliser of the action on the tree T is quasi-convex with respect to any biautomatic structure (Proposition 6.3).
In Section 7 we construct Γ; an explicit example of a non-residually finite irreducible uniform lattice in PSL 2 pRq ˆT24 as an HNN-extension.The key tool is the arithmetic of quaternion algebras which allow us to ensure the stable letter acts on RH 2 as an infinite order elliptic isometry that commensurates the vertex group.We show that the translation lengths on RH 2 of some elements of a vertex stabiliser in the tree are not rational multiples of each other (Lemma 7.3).
In Section 8 we prove Theorem A. In the appendix (Appendix A) we detail a presentation of Γ.
Acknowledgements.The authors would like to thank Ian Leary and Ashot Minasyan to whom we both owe a great intellectual debt and whose work has been a continued source of inspiration.The authors are grateful to Thomas Haettel, Mark Hagen and Ashot Minasyan for their comments on an earlier version of this paper.The first author would once again like to thank Ian Leary for his tireless, kind, and often humorous encouragement during my PhD.The first author was supported by the Engineering and Physical Sciences Research Council grant number 2127970 and the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No. 850930).Finally, we would like to thank the anonymous referee for their helpful comments.

Preliminaries
2.1.Lattices and graphs of groups.Our example will be constructed as a lattice in PSL 2 pRqˆT 24 .To this end we record some definitions and results we will use in Section 6 and Section 7.
Definition 2.1.Let H be a locally compact topological group with right invariant Haar measure µ.A discrete subgroup Γ ď H is a lattice if the covolume µpH{Γq is finite.A lattice is uniform if H{Γ is compact and nonuniform otherwise.Let S be a right H-set such that for all s P S, the stabilisers H s are compact and open, then if Γ ď H is discrete the stabilisers in the action of Γ on S are finite.
Let X be a locally finite, connected, simply connected simplicial complex.The group H " AutpXq of simplicial automorphisms of X naturally has the structure of a locally compact topological group, where the topology is given by uniform convergence on compacta.
Note that T the automorphism group of a locally-finite tree T admits lattices if and only if the group T is unimodular (that is the left and right Haar measures coincide).In this case we say T is unimodular.We say a tree T is leafless if it has no vertices of valence one.
Two notions of irreducibility for a lattice will feature in this paper.
Definition 2.2.Let T be a locally-finite unimodular leafless tree not isometric to R and let T " AutpT q be non-discrete and cocompact.Let Γ be a uniform pPSL 2 pRq ˆT q-lattice.We say that Γ is weakly irreducible if one (and hence both-see [Hug21b, Proposition 3.4]) of the images of the projections π PSL 2 pRq : Γ Ñ PSL 2 pRq and π T : Γ Ñ T are non-discrete.We say Γ is algebraically irreducible if there is no finite index subgroup Γ 1 ˆΓ2 of Γ with Γ 1 and Γ 2 infinite.By [CM09, Theorem 4.2], the two notions of irreducibility are equivalent for a pPSL 2 pRq ˆT q-lattice Γ.So if Γ is either (and hence both) weakly or algebraically irreducible we will simply state that Γ is irreducible.
To construct and study lattices in product with a tree we will utilise the graph of lattices construction from [Hug21b].Before we do this we will define graphs of groups following Bass [Bas93].
Definition 2.3.A graph of groups pA, Aq consists of a connected graph A together with some extra data A " pV A, EA, ΦAq.This data consists of vertex groups A v P V A for each vertex v, edge groups A e " A e P EA for each (oriented) edge e, and monomorphisms pα e : A e Ñ A ιpeq q P ΦA for every oriented edge in A. We will often refer to the vertex and edge groups as local groups and the monomorphisms as structure maps.
Definition 2.4.The path group πpAq has generators the vertex groups A v and elements t e for each edge e P EA along with the relations: The relations in the groups A v , t e " t ´1 e , t e α e pgqt ´1 e " α e pgq for all e P EA and g P A e " A e .
-Definition 2.5.We will often abuse notation and write A for a graph of groups.The fundamental group of a graph of groups can be defined in two ways.Firstly, considering reduced loops based at a vertex v in the graph of groups, in this case the fundamental group is denoted π 1 pA, vq (see [Bas93,Definition 1.15]).Secondly, with respect to a maximal or spanning tree of the graph.Let X be a spanning tree for A, we define π 1 pA, Xq to be the group generated by the vertex groups A v and elements t e for each edge e P EA with the relations: The relations in the groups A v , t e " t ´1 e for each (oriented) edge e, t e α e pgqt ´1 e " α e pgq for all g P A e , t e " 1 if e is an edge in X.
, / / ./ / -Note that the definitions are independent of the choice of basepoint v and spanning tree X and both definitions yield isomorphic groups so we can talk about the fundamental group of A, denoted π 1 pAq.
We say a group G is covirtually isomorphic to H if there exists a finite normal subgroup N Ĳ G such that G{N -H.We are now ready to define a graph of PSL 2 pRq-lattices.
Definition 2.6.A graph of PSL 2 pRq-lattices pA, A, ψq is a graph of groups pA, Aq that is equipped with a morphism of graphs of groups ψ : A Ñ PSL 2 pRq such that: (1) Each local group A σ P A is covirtually a PSL 2 pRq-lattice and the image ψpA σ q is a PSL 2 pRq-lattice; (2) The local groups are commensurable in Γ " π 1 pAq and their images are commensurable in PSL 2 pRq; (3) For each e P EA the element t e of the path group πpAq is mapped under ψ to an element of Comm PSL 2 pRq pψ e pA e qq.
The relevance of a graph of PSL 2 pRq-lattices is the following special case of [Hug21b, Theorem A].
Theorem 2.7.[Hug21b, Theorem A] Let pA, A, ψq be a finite graph of PSL 2 pRq-lattices with locally-finite unimodular non-discrete Bass-Serre tree T , and fundamental group Γ. Suppose T " AutpT q admits a uniform lattice.If each local group A σ is covirtually a uniform PSL 2 pRq-lattice, and the kernel Kerpψ| Aσ q acts faithfully on T , then Γ is a uniform pPSL 2 pRqˆT q-lattice and hence a CATp0q group.Conversely, if Λ is a uniform pPSL 2 pRq ˆT q-lattice, then Λ splits as a finite graph of uniform PSL 2 pRq-lattices with Bass-Serre tree T .
2.2.Biautomatic structures.We are interested in studying when a group G is biautomatic; we briefly introduce the necessary definitions and basic results on biautomaticity below, and refer the interested reader to [Eps+92] for a more comprehensive account.
We remark that the nowadays standard definition of a biautomatic structure that we give below differs from [Eps+92, Definition 2.5.4](see [Amr21] for an explanation).However, for finite-to-one structures these definitions are equivalent [Amr21, Theorem 6].
Let G be a group with a finite generating set A. Formally, we view A as a finite set together with a function π 0 A : A Ñ G that extends to a surjective monoid homomorphism π A : A ˚Ñ G, where A ˚is the free monoid on A; we say that a word v P A ˚labels the element π A pvq P G.For simplicity, we will assume that A is symmetric (π A pAq " π A pAq ´1) and contains the identity (π A p1q " 1 G for an element 1 P A).We denote by d A the combinatorial metric on the Cayley graph CaypG, Aq of G.
We study (combinatorial) paths in CaypG, Aq.Given a path p in CaypG, Aq and an integer t P t0, . . ., |p|u, where |p| is the length of p, we denote by p pptq P G the t-th vertex of p, so that p pp0q and p pp|p|q are the starting and ending vertices of p, respectively.We further define p pptq P G for any t P Z ě0 Y t8u by setting p pptq " p pp|p|q whenever t ą |p|.
Definition 2.8.Let G be a group with a finite symmetric generating set A containing the identity, and let L Ď A ˚. We say pA, Lq is a (uniformly finite-to-one) biautomatic structure on G if (i) L is recognised by a finite state automaton over A; (ii) there exists N ě 1 such that 1 ď |π ´1 A pgq X L| ď N for every g P G; and (iii) L satisfies the "two-sided fellow traveller property": there exists a constant ζ ě 1 such that if p and q are paths in CaypG, Aq labelled by words in L and satisfying d A pp pp0q, p qp0qq ď 1 and d A pp pp8q, p qp8qq ď 1, then d A pp pptq, p qptqq ď ζ for all t.We say G is biautomatic if it has some uniformly finite-to-one biautomatic structure.
The standard notion of a biautomatic structure appearing in the literature (cf [Eps+92]) is more general than the notion of a uniformly finite-to-one biautomatic structure as defined here.Nevertheless, it can be shown that every biautomatic group (in the sense of [Eps+92], for instance) has a uniformly finite-to-one biautomatic structure [Eps+92, Theorem 2.5.1] and so is biautomatic in our sense as well.In this paper, we assume all biautomatic structures to be uniformly finite-to-one.
We record the following result for future reference; for part (i), it is enough to take ν to be larger than the number of states in an automaton over A recognising L.
Theorem 2.9 (D.B. A. Epstein et al. [Eps+92, Lemma 2.3.9 & Theorem 3.3.4]).Let pA, Lq be a biautomatic structure on a group G. Then there exists a constant ν ě 1 with the following properties: (i) if v P A ˚is a subword of a word w P L, then there exist u 1 , u 2 P A such that |u 1 |, |u 2 | ď ν and u 1 vu 2 P L, and if v is a prefix (respectively suffix) of w, then we can take u 1 " 1 (respectively u 2 " 1); (ii) if v, w P L are such that π A pvq " π A pwaq or π A pvq " π A pawq for some a P A, then ˇˇ|v| ´|w| ˇˇď ν; and (iii) any path in CaypG, Aq labelled by a word in L is a pν, νq-quasigeodesic.
The following notion will be crucial in our arguments.
Definition 2.10.Let pA, Lq be a biautomatic structure on a group G, and let H Ď G.We say that H is L-quasiconvex if there exists a constant ξ ě 1 such that every path in CaypG, Aq starting and ending at vertices of H and labelled by a word in L belongs to the ξ-neighbourhood of H.
The importance of the notion of L-quasiconvexity can be summarised in the following result.It can be extracted from the proofs of [GS91, Theorem 3.1 & Proposition 4.3] and from [Eps+92, Lemma 2.3.9].(i) For any g 1 , . . ., g n P G, the centraliser C G ptg 1 , . . ., g n uq is M-quasiconvex.
(ii) Let H ď G be an M-quasiconvex subgroup.Then there exists a biautomatic structure pA, Lq on H and a constant κ ě 1 such that if v P M and w P L represent the same element of G, then ˇˇ|v| ´|w| ˇˇď κ.
A biautomatic structure pA, Lq on H ď G appearing in Theorem 2.11(ii) will be called a biautomatic structure associated to pB, Mq.
Finally, we record the following observation.
Lemma 2.12.Let pA, Lq be a biautomatic structure on a group G, let H 1 ď H 2 ď G, and suppose that rH 2 : Proof.Note that since rH 2 : H 1 s ă 8, there exists a constant λ ě 1 such that H 2 belongs to the λ-neighbourhood of H 1 in CaypG, Aq.Moreover, let ζ ě 1 be the constant appearing in Definition 2.8(iii).Suppose first that H 1 is L-quasiconvex, and let ξ 1 ě 1 be the constant appearing in Definition 2.10.Let p 2 be a path in CaypG, Aq labelled by a word in L with p p 2 p0q, p p 2 p8q P H 2 .Since H 2 belongs to the λ-neighbourhood of H 1 , there exist g ´, g `P H 1 such that d A pg ´, p p 2 p0qq ď λ and d A pg `, p p 2 p8qq ď λ; moreover, since π A | L is surjective, there exists a path p 1 in CaypG, Aq labelled by a word in L, starting at g ´and ending at g `.It then follows that p 2 is in the λζ-neighbourhood of p 1 , and p 1 is in the ξ 1 -neighbourhood of H 1 .Therefore, p 2 is in the pλζ `ξ1 q-neighbourhood of H 1 , and so of H 2 ; it follows that H 2 is L-quasiconvex, as required.
Conversely, suppose that H 2 is L-quasiconvex, and let ξ 2 ě 1 be the constant appearing in Definition 2.10.Then any path in CaypG, Aq labelled by a word in L and with endpoints in H 1 belongs to the ξ 2 -neighbourhood of H 2 , and so to the pξ 2 `λq-neighbourhood of H 1 .It follows that H 1 is L-quasiconvex, as required.
2.3.Geodesic currents.We now fix a closed orientable Riemannian surface Σ of constant curvature ´1, and let G " π 1 pΣq.We also fix the universal covering map r Σ Ñ Σ and the G-action by isometries on r Σ.Let I `p r Σq be the set of oriented (i.e.directed) geodesic lines on r Σ.Since each such geodesic line is uniquely determined by its endpoints on B r Σ -S 1 , we can topologise Σq by identifying it with the open cylinder tpx, yq P S 1 ˆS1 | x ‰ yu.Note that the G-action on r Σ induces an action of G on I `p r Σq.By an (oriented ) curve on Σ we mean a free homotopy class of essential continuous maps S 1 Ñ Σ.We denote by C `pΣq the set of all curves on Σ, which can also be identified with the set of non-trivial G-conjugacy classes.Given a primitive curve γ P C `pΣq (meaning that γ ‰ η n for any η P C `pΣq and n ě 2), we may associate a Borel measure µ γ on I `p r Σq as follows: let p γ : S 1 Ñ Σ be the unique (up to reparametrisation of S 1 ) geodesic representative of γ, let Apγq Ă I `p r Σq be the set of all lifts of p γ, and let µ γ pEq :" |E X Apγq| for any Borel subset E Ď I `p r Σq.We may also define this when γ is not primitive, by setting µ η n :" nµ η for primitive η P C `pΣq and n ě 2. By construction, Apγq, and so µ γ , is G-invariant; moreover, one can see that Apγq is discrete in I `pΣq, implying that µ γ is a Radon measure.This motivates the following definition.Definition 2.13.An (oriented ) geodesic current on Σ is a G-invariant Radon measure on I `p r Σq.The set of all geodesic currents on Σ form a topological space G `pΣq under the weak* topology: we have µ n Ñ µ in G `pΣq if and only if ş f dµ n Ñ ş f dµ for all continuous functions f : I `p r Σq Ñ R with compact support.By slightly abusing the notation, we will identify a curve γ P C `pΣq with the corresponding geodesic current γ :" µ γ P G `pΣq, and will therefore view C `pΣq as a subset of G `pΣq.
It is known that a current µ is uniquely determined by the values of ş f dµ for compactly supported continuous functions f : I `p r Σq Ñ R, as a consequence of the following theorem.
Theorem 2.14 (Riesz Representation Theorem; see [Mar20, Theorem 1.7.13]).Let X be a locally compact Hausdorff space, and let C c pXq be the set of continuous functions f : X Ñ R with compact support.For any linear functional F : C c pXq Ñ R such that F pf q ě 0 whenever f pxq ě 0 for all x P X, there exists a unique Radon measure µ on X such that F pf q " ş f dµ for all f P C c pXq.In particular, if µ and µ 1 are Radon measures on X such that ş f dµ " ş f dµ 1 for all f P C c pXq, then µ " µ 1 .
A core part of this paper is based on studying certain functions f : C `pΣq Ñ R. The terminology we use below roughly follows the terminology of [MT21] and [Mar20]; however, since the functions we consider are assumed to satisfy the additive union property in the sense of [MT21, Definition 1.1], we are able to make some simplifications to the statements of results.Given two maps p γ 1 , p γ 2 : S 1 Ñ Σ, a crossing of p γ 1 and p γ 2 is a pair px 1 , x 2 q P S 1 ˆS1 such that p γ 1 px 1 q " p γ 2 px 2 q, and a self-crossing of p γ : S 1 Ñ Σ is a pair px 1 , x 2 q P S 1 ˆS1 such that x 1 ‰ x 2 and p γpx 1 q " p γpx 2 q.A crossing or a self-crossing is essential if, roughly speaking, it is unavoidable in a homotopy class: see [ (i) We say f is homogeneous if f pγ n q " nf pγq for all γ P C `pΣq and n ě 1. (ii) We say f satisfies the join quasi-smoothing property if there exists a constant ζ ě 0 such that the following holds.Let px 1 , x 2 q be an essential crossing of maps p γ 1 , p γ 2 : S 1 Ñ Σ representing curves γ 1 , γ 2 P C `pΣq, respectively, and let γ P C `pΣq be the homotopy class of a curve obtained by cutting p γ i at x i and regluing the four resulting endpoints in a way that respects the orientation of the p γ i .Then f pγq ď f pγ 1 q `f pγ 2 q `ζ.(iii) We say f satisfies the split quasi-smoothing property if there exists a constant ζ ě 0 such that the following holds.Let px 1 , x 2 q be an essential self-crossing of a map p γ : S 1 Ñ Σ representing a curve γ P C `pΣq, and let γ 1 , γ 2 P C `pΣq be the homotopy classes of the two curves obtained by cutting p γ at x 1 and x 2 and regluing the four resulting endpoints in a way that respects the orientation of p γ. Then f pγ 1 q `f pγ 2 q ď f pγq `ζ.
Given a function f : C `pΣq Ñ R that satisfies the join and split quasismoothing properties, the following result allows us to construct such a function that is also homogeneous.
Theorem 2.16 (D.Martínez-Granado and D. P. Thurston [MT21, Theorem B]).Let f : C `pΣq Ñ R be a function satisfying the join and split quasi-smoothing properties.Then the function f : C `pΣq Ñ R defined by f pγq " lim nÑ8 f pγ n q{n is well-defined, homogeneous, and satisfies the join and split quasi-smoothing properties.
The main motivation for these definitions arises from the following result that is crucial in our argument.
Theorem 2.17 (D.Martínez-Granado and D. P. Thurston [MT21, Theorem A]).Let f : C `pΣq Ñ R be a homogeneous function satisfying the join and split quasi-smoothing properties.Then f extends to a unique continuous homogeneous function f : G `pΣq Ñ R.
As a consequence of Theorems 2.16 and 2.17, if a function f : C `pΣq Ñ R satisfies the join and split quasi-smoothing properties, then f : C `pΣq Ñ R extends to a unique continuous homogeneous function f : G `pΣq Ñ R.
Another property we will use is "positive linearity".We say a function Lemma 2.18.Let f : C `pΣq Ñ R be a homogeneous function satisfying the join and split quasi-smoothing properties.Then the function f : G `pΣq Ñ R given by Theorem 2.17 is positively linear.
Proof.Let R `C`p Σq Ă G `pΣq be the subspace of currents of the form ř i c i γ i for some c i ě 0 and γ i P C `pΣq.Since f : C `pΣq Ñ R is homogeneous, we can extend it to a function p f : R `C`p Σq Ñ R by setting p f p ř i c i γ i q :" ř i c i f pγ i q.The fact that f : C `pΣq Ñ R satisfies the join and split quasi-smoothing properties in our terminology implies that p f : R `C`p Σq Ñ R satisfies quasismoothing in the terminology of [MT21].In particular, by the uniqueness in Theorem 2.17, the restriction of f : 2.4.Intersection numbers.Finally, we study the intersection numbers between currents.Let DI `p r Σq Ă I `p r Σq ˆI`p r Σq be the set of pairs pγ 1 , γ 2 q of geodesic lines on r Σ that intersect transversely; one can show that DI `p r Σq is a 4-manifold.The G-action on r Σ induces a free and properly discontinuous action on DI `p r Σq, and so we may define the quotient DI `pΣq :" DI `p r Σq{G.
We say λ P G `pΣq is a filling current if every geodesic line in r Σ transversely intersects another geodesic line contained in the support of λ.
We record the following observation on intersection numbers for future reference.
Proof.The isometry r ϕ induces a homeomorphism I `p r Σ 1 qˆI `p r Σ 1 q Ñ I `p r ΣqÎ `p r Σq that maps DI `p r Σ 1 q onto DI `p r Σq; let r ϕ 1 : DI `p r Σ 1 q Ñ DI `p r Σq be this induced map.Moreover, we can canonically identify DI `pΣq with the set of triples px, t 1 , t 2 q, where x P Σ and t 1 , t 2 P T 1 x Σ -S 1 are such that t 1 ‰ t 2 , and the topology is the "usual" one (see [Bon86]); this viewpoint allows us to see that ϕ induces a k-sheeted covering map ϕ 1 : DI `pΣ 1 q Ñ DI `pΣq.One may check that we have p Σ ˝r ϕ 1 " ϕ 1 ˝pΣ 1 , where p Σ : DI `p r Σq Ñ DI `pΣq and p Σ 1 : DI `p r Σ 1 q Ñ DI `pΣ 1 q are the canonical covering maps.It follows that pµ 1 ˝r ϕqbpµ 2 ˝r ϕq " pµ 1 bµ 2 q˝ϕ 1 , i.e. we have rpµ 1 ˝r ϕq b pµ 2 ˝r ϕqs pAq " pµ 1 b µ 2 qpϕ 1 pAqq for every Borel subset A Ď DI `pΣ 1 q such that ϕ 1 | A is injective.This implies that rpµ 1 ˝r ϕq b pµ 2 ˝r ϕqs pDI `pΣ 1 qq " k ¨pµ 1 b µ 2 qpDI `pΣqq, as required.

Stable word lengths
Throughout this section, we fix a biautomatic group G with a (uniformly finite-to-one) biautomatic structure pA, Lq.We define several functions G Ñ R associated to lengths of words in L, and study the relationship between them.
Proposition 3.1.Let g P G, and for each n ě 1 let w n P L be a word representing g n .Then the sequence p|w n |{nq 8 n"1 converges and the limit lim nÑ8 |w n |{n is rational.
Proof.Suppose first that g has finite order, and so the subgroup xgy is finite.Since pA, Lq is finite-to-one, it follows that the set tw n | n ě 1u is finite, and so the sequence p|w n |q 8 n"1 is bounded.Therefore, |w n |{n Ñ 0 P Q as n Ñ 8, which implies the result.
Suppose now that g has infinite order.By Theorem 2.11(i), the centraliser C G pgq is L-quasiconvex (with associated structure pA 1 , L 1 q, say), and so finitely generated, implying (again by Theorem 2.11) that its centre ZpC G pgqq is L 1 -quasiconvex (with associated structure pA 2 , L 2 q, say).Thus ZpC G pgqq is a finitely generated abelian group containing g, and so (as g has infinite order) we have ZpC G pgqq " H ˆF , where H -Z N , g P H, and F is finite.In particular, H has finite index in ZpC G pgqq, and is therefore L 2 -quasiconvex; let pB, Mq be the associated biautomatic structure on H.
By applying Theorem 2.11(ii) three times, it follows that there exists a constant κ ě 1 such that for each n ě 1, if v n P M is a word representing g n then ˇˇ|v n | ´|w n | ˇˇď κ.Moreover, since pB, Mq is a biautomatic structure on H -Z N , there exists a constant ρ ě 0 and a function f : H Ñ Q such that f ph n q " nf phq for all h P H and n ě 1, and such that ˇˇf phq ´|u| ˇˇď ρ for any h P H and any word u P M representing h [Val21b, Proposition 4.2 and its proof].In particular, for each n ě 1 we have and so |w n |{n Ñ f pgq P Q as n Ñ 8, as required.
Motivated by this, we introduce the following terminology.
(i) The L-word length of g, denoted |g| L , is the length of the shortest word in L representing g. (ii) The conjugacy L-word length of g is defined as }g} L :" min hPG ˇˇhgh ´1ˇL .
(iii) The stable L-word length of g is defined as τ L pgq :" lim nÑ8 |g n | L {n.
It follows from Proposition 3.1 that τ L pgq is well-defined: indeed, this number is equal to lim nÑ8 |wn| n in the notation of the Proposition.We make the following easy observation.
Proof.Let ν ě 1 be the constant given in Theorem 2.9, and let w n , v n P L be the shortest words representing g n , hg n h ´1, respectively.Then ˇˇ|w n | |v n | ˇˇď 2ν maxt|h| A , 1u, and so τ L pgq " lim nÑ8 |w n |{n " lim nÑ8 |v n |{n " τ L phgh ´1q, as required.
In particular, it follows that τ L pgq " min hPG τ L phgh ´1q " min hPG lim nÑ8 ˇˇhg n h ´1ˇL {n for any g P G.In the remainder of this section, we prove that if G is hyperbolic then the minimum and the limit in this expression can be swapped, and therefore τ L pgq " lim nÑ8 }g n } L {n for all g P G.
Lemma 3.4.Suppose G is hyperbolic.Then there exist constants λ ě 1 and ε ě 0 satisfying the following property.Let g P G be an element of infinite order, and let w P L be a word representing a conjugate of g with The proof of Lemma 3.4.The thick path is γ, the red subpath is η, and the blue and green paths are labelled by words in L.
Proof.Let ν ě 1 be the constant given in Theorem 2.9.Then there exist constants , λ ě 1 and ε ě 0 such that every -local pν, 2νpν `1qq-quasi-geodesic in CaypG, Aq is a pλ, εq-quasi-geodesic [CDP90, Chapitre 3, Théorème 1.4].Moreover, if w P A ˚is a word representing an infinite order element, then any bi-infinite path in CaypG, Aq labelled by ¨¨¨www ¨¨¨is a quasi-geodesic [GH90, Proposition 8.21].Since A is finite, there are only finitely many words w P L of length ă ; therefore, after increasing λ ě 1 and ε ě 0 if necessary, we may assume that for every word w P L with |w| ă representing an element of infinite order, a path in CaypG, Aq labelled by ¨¨¨www ¨¨¨is a pλ, εq-quasi-geodesic.
It is therefore enough to show the following: if g P G and w P L are such that w represents g and |w| " }g} L ě , and if γ Ď CaypG, Aq is a bi-infinite path labelled by ¨¨¨www ¨¨¨, then any subpath of γ of length ď is a pν, 2νpν `1qq-quasi-geodesic.Thus, let η Ă γ be a subpath of length ď from h P γ to k P γ.We aim to show that |η| ď νd A ph, kq `2νpν `1q.
Since |η| ď ď |w|, it follows that η is labelled by a subword of ww; however, since paths labelled by w are pν, νq-quasi-geodesic, we may without loss of generality assume that η is not labelled by subword of w.Thus η is labelled by a word w 1 w 2 , where w 1 and w 2 are a suffix and a prefix of w, respectively.Since |w 1 | `|w 2 | " |η| ď ď |w|, it follows that w " w 2 vw 1 for some v P A ˚. Since v is a subword of a word in L, there exist words u 1 , u 2 P A ˚of length ď ν such that u 1 vu 2 P L. See Figure 1.Now let g 1 P G be the element represented by w 1 w 2 v P A ˚, so that g and g 1 are conjugate in G, and let w 1 P L be a word representing g 1 .By construction, we have w 1 , u 1 vu 2 P L, and there exist paths in It follows that |η| ď νd A ph, kq `2νpν `1q, as required.
. The proof of Lemma 3.5.
Lemma 3.5.Suppose G is hyperbolic.Then }g n } L {n Ñ τ L pgq as n Ñ 8 for every g P G.
Proof.Fix g P G.If g has finite order, then the set t}g n } L | n ě 1u is bounded, implying that }g n } L {n Ñ 0 " τ L pgq as n Ñ 8, as required.Therefore, we may without loss of generality assume that g has infinite order.After replacing g by its conjugate if necessary (we can do this by Lemma 3.3), we may assume }g} L " |w 1 | for some word w 1 P L representing g.
For each n ě 1, let h n P G and w n P L be such that w n represents h ´1 n g n h n and }g n } L " |w n |.By replacing h n with h n g M for some M " M pnq P Z if necessary, we may assume that d A ph n , 1 G q ď d A ph n , g m q for all m P Z (when n is fixed); in particular, we may take h 1 " 1 G .Let γ n be a bi-infinite path in CaypG, Aq labelled by ¨¨¨w n w n w n ¨¨¨such that a sub-ray of γ n labelled by w n w n w n ¨¨¨starts at the vertex h n .
By Lemma 3.4, there exist constants λ ě 1 and ε ě 0 such that each γ n is a pλ, εq-quasi-geodesic.Furthermore, since γ n and γ 1 contain vertices g mn h n and g mn , respectively, for all m P Z, it follows that γ n and γ 1 have the same endpoints on the boundary BG.Therefore, by [CDP90, Chapitre 3, Théorème 3.1], there exists a constant β ě 0 such that γ n is Hausdorff distance ď β away from γ 1 for each n P Z. See Figure 2. Now since h n P γ n , there exists a vertex k n P γ 1 such that d A ph n , k n q ď β.Since γ 1 is the union of the xgy-translates of a path labelled by w 1 and starting at 1 G , we have d A pg M , k n q ď |w 1 | for some M " M pnq P Z.But then the minimality of As A is finite, it follows that the set C :" th n | n ě 1u is finite.Now for each h P C and n ě 1, let v n,h P L be a word representing h ´1g n h P G.We then have }g n } L " mint|v n,h | | h P Cu.Furthermore, it follows from Proposition 3.1 that |v n,h |{n Ñ τ L phgh ´1q and therefore, by Lemma 3.3, |v n,h |{n Ñ τ L pgq as n Ñ 8, for each h P C. As C is finite, we thus have }g n } L {n Ñ τ L pgq as n Ñ 8, as required.

Quasi-smoothing
Throughout this section, we fix a hyperbolic group G together with a (uniformly finite-to-one) biautomatic structure pA, Lq on G.We use the notation of Definition 3.2.Lemma 4.1.Let G be a hyperbolic group, and let pA, Lq be a finite-to-one biautomatic structure on G. Then there exists a constant ξ ě 0 such that the following hold: Let w 1 , w 2 , v 1 1 , v 1 2 P A ˚be the labels of γ 1 , γ 2 , ζ 11 , ζ 22 , respectively.Then there exist words u 1 , u 2 , t 1 , t 2 P A ˚, all of length ď ν, such that w 1 u 1 , u 2 w 2 , v 1 1 t 1 , t 2 v 1 2 P L; see Figure 3a.It follows that the endpoints of the paths starting at 1 G and labelled by w 1 u 1 and by as required.Now let Σ be a closed orientable hyperbolic surface, let G " π 1 pΣq, and let pA, Lq be a biautomatic structure on G as before.We may then identify C `pΣq with the set of non-trivial conjugacy classes in G. Since the function }´} L : G Ñ R is by definition invariant under conjugacy in G, it factors through a function C `pΣq Ñ R which we also denote by }´} L .We aim to show that }´} L : C `pΣq Ñ R satisfies the join and split quasi-smoothing properties: see Definition 2.15.
Proposition 4.2.The function }´} L satisfies the join and split quasi-smoothing properties.
Proof.Let V " CaypG, Aq{G: that is, V is a rose-a graph with one vertexwith one loop edge for each element of A. We will not distinguish pointed loops in V from their pointed homotopy classes, allowing us to assign to each such loop a label w P A ˚. Let π V : CaypG, Aq Ñ V and π Σ : r Σ Ñ Σ be the canonical covering maps, and let θ : V Ñ Σ be a continuous map that sends each edge in V to a pointed loop on Σ labelled by the corresponding element of A Ă G " π 1 pΣq.
Since θ ˝πV maps loops in CaypG, Aq to nullhomotopic loops in Σ and so induces a trivial map π 1 pCaypG, Aqq Ñ π 1 pΣq, it follows that θ ˝πV " π Σ ˝r θ for some map r θ : CaypG, Aq Ñ r Σ.Moreover, r θ is clearly G-equivariant; since V and Σ are both compact and the G-action on r Σ is properly discontinuous, it follows by the Švarc-Milnor Lemma that r θ is a pλ, εq-quasi-isometry for some λ ě 1 and ε ě 0, implying that the diameter of r θ ´1pr xq is at most λε for any r x P r Σ.In particular, if x 1 , x 2 P V are such that θpx 1 q " θpx 2 q, then there are lifts r x 1 P π ´1 V px 1 q and r x 2 P π ´1 V px 2 q such that r θpr x 1 q " r θpr x 2 q.This implies that d CaypG,Aq pr x 1 , r x 2 q ď λε; therefore, if r γ : r0, 1s Ñ CaypG, Aq is a geodesic from r x 1 to r x 2 , then γ :" π V ˝r γ is a path in V of length ď λε that is mapped (under θ) to a nullhomotopic loop on Σ.
Let ν be the constant given by Theorem 2.9, and let β ě 0 be the constant such that any two pν, νq-quasi-geodesic paths in CaypG, Aq with the same endpoints are Hausdorff distance ď β apart: such a β exists by [CDP90, Chapitre 3, Théorème 1.2].We set where ξ is the constant given in Lemma 4.1.We now prove the (i) join quasi-smoothing and (ii) split quasi-smoothing properties.
(i) For i P t1, 2u, let γ i P C `pΣq, and let w i P L represent an element in the conjugacy class corresponding to γ i such that |w i | " }γ i } L ; moreover, let σ i : S 1 Ñ V be the (pointed) loop on V labelled by w i , so that the loop p γ i :" θ ˝σi is in the free homotopy class γ i .Suppose py 1 , y 2 q is an essential crossing of p γ 1 and p γ 2 , so that p γ 1 py 1 q " p γ 2 py 2 q, and let γ P C `pΣq be the path obtained by the join quasi-smoothing procedure as in Definition 2.15.We can thus write σ 1 " σ 11 ¨σ12 and σ 2 " σ 21 ¨σ22 for some σ ij : r0, 1s Ñ V , where we write σ 1 ¨σ2 for concatenation of paths σ 1 and σ 2 (under some reparametrisation), so that pθ ˝σ11 q ¨pθ ˝σ22 q ¨pθ ˝σ21 q ¨pθ ˝σ12 q is in the homotopy class γ.
Let x i :" σ i py i q P V for i P t1, 2u, so that θpx 1 q " θpx 2 q.Then, as explained above, there exists a path η : r0, 1s Ñ V from x 1 to x 2 of length ď λε such that the loop θ ˝η is nullhomotopic in Σ.It follows that θ ˝pσ 11 ¨η ¨σ22 ¨σ21 ¨η ¨σ12 q is a well-defined loop that is in the homotopy class γ, where η : r0, 1s Ñ V can be taken to be the "reverse" of η.See Figure 4a.
By adding or removing initial and terminal subpaths of length at most one to/from the paths σ ij , η and η, we may modify our construction so that each of these paths start and end at the vertex of V .In particular, there exist paths σ 1 11 , σ 1 12 , σ 1 21 , σ 1 22 , η 1 , η 1 : r0, 1s Ñ V , all starting and ending at the vertex of V , such that σ 1 " σ 1 12 is a well-defined loop that is mapped under θ to the homotopy class γ.
Let w 11 , w 12 , w 21 , w 22 , v, v P A ˚be the labels of the paths σ 1 11 , σ 1 12 , σ 1 21 , σ 1 22 , η 1 , η 1 , respectively.We then have w 1 " w 11 w 12 P L, w 2 " w 21 w 22 P L, and |v|, |v| ď λε `2; moreover, the G-conjugacy class of w 11 vw 22 w 21 vw 12 corresponds to the homotopy class γ.If, given u P A ˚, we write |u| L for |g| L , where g P G is the element represented by u, then Lemma 4.1 implies that as required.(ii) Let γ P C `pΣq, and let w P L represent an element in the conjugacy class corresponding to γ such that |w| " }γ} L ; moreover, let σ : S 1 Ñ V be the (pointed) loop on V labelled by w, so that the loop p γ :" θ ˝σ is in the free homotopy class γ.Suppose py 1 , y 2 q is an essential self-crossing of p γ, and let γ 1 , γ 2 P C `pΣq be the paths obtained by the split quasi-smoothing procedure as in Definition 2.15.Similarly to the previous case (see Figure 4b), we may find paths σ 1 1 , σ 1 2 , σ 1 3 , η 1 , η 1 : r0, 1s Ñ V , all starting and ending at the vertex of 3 , such that σ 1 1 ¨η1 ¨σ1 3 and σ 1 2 ¨η1 are well-defined loops that are mapped (under θ) to the free homotopy classes γ 1 and γ 2 , respectively, and such that η 1 and η 1 have length ď λε `2.
Let w 1 , w 2 , w 3 , v, v P A ˚be the labels of the paths σ 1 1 , σ 1 2 , σ 1 3 , η 1 , η 1 , respectively.It then follows that w " w 1 w 2 w 3 P L, that |v|, |v| ď λε`2, and that the G-conjugacy classes of w 1 vw 3 and w 2 v correspond to the homotopy classes γ 1 and γ 2 , respectively.Now let u P L be a word such that u and w 1 w 2 represent the same element of G. Since u and w 1 w 2 are both pν, νq-quasi-geodesic words, we can write u " u 1 u 2 so that u 1 s and w 1 represent the same element of G for some s P A ˚with |s| ď β; consequently, s ´1u 2 and w 2 also represent the same element of G.We then have as required.

Invariant measures
In this section, we fix a closed orientable hyperbolic surface Σ and let G " π 1 pΣq.We view G as a uniform lattice in PSL 2 pRq -Isom `p r Σq, the group of orientation-preserving isometries of the universal cover r Σ -RH 2 of Σ.
Note that PSL 2 pRq acts smoothly, freely and transitively on T 1 r Σ, the unit tangent bundle of r Σ.We thus have a diffeomorphism PSL 2 pRq -T 1 r Σ.Under this diffeomorphism, the G-action on T 1 r Σ corresponds to the G-action on PSL 2 pRq by left multiplication, and the R-action on T 1 r Σ by translations along lifts of geodesic lines on r Σ corresponds to the action of a subgroup R -R on PSL 2 pRq by right multiplication; see [Mar20, §1.8.3].After noticing that PSL 2 pRq{R -I `p r Σq, we can then identify G `pΣq with a certain space of measures on PSL 2 pRq, as follows.
Proposition 5.1 (Y.Benoist and H. Oh [BO07, Proposition 8.1]).Let G 1 pΣq be the space of Radon measures on PSL 2 pRq that are G-invariant on the left and R-invariant on the right, equipped with the weak* topology.Then the map where µ 1 pEq " ş λ R pg ´1E X Rq dµpgRq for a Borel subset E Ď PSL 2 pRq and λ R is a left Haar measure on R, is a homeomorphism.θpσ 21 q θpσ 12 q θpσ 11 q (a) Join quasi-smoothing, (i). is the one at which the quasi-smoothing procedure is done, and the blue paths have length ď λε.
Throughout this section, we will thus identify G `pΣq with the space G 1 pΣq in Proposition 5.1.We will assume all the measures on PSL 2 pRq in this section to be R-invariant on the right.We will also fix a left Haar measure λ Σ on PSL 2 pRq.We may rescale λ Σ so that ιpγ, λ Σ q is equal to the length of the geodesic representative p γ : S 1 Ñ Σ for any γ P C `pΣq: see [Mar20, §1.8.3].Now let µ be a Radon measure on PSL 2 pRq that is G 0 -invariant for some finite index subgroup G 0 of G.We then construct a current p µ P G `pΣq as follows.Let g 1 , . . ., g s be a right transversal of G 0 in G. Given a Borel subset E Ď PSL 2 pRq, we then set It is straightforward to check that p µ is indeed G-invariant and does not depend on the choice of the right transversal G 0 .
We consider the following special case.Let µ P G `pΣq, and let t P PSL 2 pRq be such that G 0 :" t ´1Gt XG has finite index in G. Then the measure µpt´q is G 0 -invariant.We define µ ptq :" p µ 1 , where µ 1 " µpt´q.Given an element t P PSL 2 pRq, we write x¨G, t¨y for the submonoid of PSL 2 pRq generated by G Y ttu, and x¨t¨y for the submonoid generated by t.
Lemma 5.2.Let t P PSL 2 pRq be an elliptic isometry of RH 2 .If xG, ty is dense in PSL 2 pRq, then so is x¨G, t¨y.
Proof.If t has finite order (m, say), then we have t ´1 " t m´1 P x¨t¨y and so xG, ty " x¨G, t¨y.Therefore, without loss of generality we may assume that t has infinite order.
Since t is elliptic, it stabilises a point x 0 P RH 2 .As t has infinite order, the submonoid x¨t¨y of Stab PSL 2 pRq px 0 q -S 1 is infinite, and so dense in Stab PSL 2 pRq px 0 q (by the Dirichlet's Approximation Theorem, for instance).In particular, for any open neighbourhood U Ď PSL 2 pRq of t ´1, there exists m P N such that t m P U .Now let V Ď PSL 2 pRq be open.Since xG, ty is dense, there exists h P xG, ty such that h P V .We can write h " h 0 t ´1h 1 t ´1 ¨¨¨t ´1h n for some h 0 , . . ., h n P x¨G, t¨y.Consider the map ϕ : PSL 2 pRq Ñ PSL 2 pRq defined by ϕpgq " h 0 gh 1 g ¨¨¨gh n , and note that ϕpt ´1q " h P V .Since the multiplication in PSL 2 pRq is continuous, so is the map ϕ, implying that ϕ ´1pV q is an open neighbourhood of t ´1 in PSL 2 pRq.But then t m P ϕ ´1pV q for some m P N, implying that ϕpt m q P V X x¨G, t¨y, and in particular that V X x¨G, t¨y ‰ ∅.As V was an arbitrary open subset, it follows that x¨G, t¨y is dense in PSL 2 pRq, as required.
Lemma 5.3.Let t P PSL 2 pRq be an element such that G 0 :" t ´1Gt X G has finite index in G, and such that the monoid x¨G, t¨y is dense in PSL 2 pRq.Let µ P G `pΣq be a non-zero current such that µ ptq " µ.Then µ " k ¨λΣ for some k ą 0.
Proof.We aim to show that µph´q " µ for all h P PSL 2 pRq: this will imply the result by the uniqueness of the Haar measure.
Let f : PSL 2 pRq Ñ R be a continuous function with compact support K Ă PSL 2 pRq, and consider the map I f : PSL 2 pRq Þ Ñ R given by I f pgq " ş f dµpg´q.Such a map I f is continuous: see [Gaa73, Lemma 15 on p. 278] and its proof.Now since µ is G-invariant, it follows that I f pghq " I f phq whenever g P G, implying that I f factors through the map PSL 2 pRq Ñ GzPSL 2 pRq -T 1 Σ.Since T 1 Σ is compact, so is the image of I f , and so I f attains its infimum: that is, the set M f :" tx P PSL 2 pRq | I f pxq ď I f pyq for all y P PSL 2 pRqu is non-empty.Now let g 1 , . . ., g s be a right transversal of G 0 in G with g 1 " 1.We then have µpEq " µ ptq pEq " s ´1 ř s i"1 µptg i Eq for any Borel subset E. In particular, it follows that for any x P PSL 2 pRq, Therefore, if x P M f then I f ptg i xq " I f pxq for all i.In particular, I f ptxq " I f pxq: that is, tx P M f .Thus, if x P M f , then tx P M f and gx P M f for all g P G, implying that x¨G, t¨y M f Ď M f .As x¨G, t¨y is dense in PSL 2 pRq and M f ‰ ∅, it follows that M f is also dense; as I f is continuous, this implies that I f is actually constant on PSL 2 pRq.But since f was arbitrary, it follows from Theorem 2.14 that µph´q " µ for all h P PSL 2 pRq, as required.
Proof.Let g 1 , . . ., g s be a right transversal of G 0 in G with g 1 " 1.For 1 ď i ď s, let Σ i Ñ Σ be the finite covering map corresponding to the subgroup g ´1 i G 0 g i ď G, and let Σ 0 Ñ Σ be the finite covering map corresponding to the subgroup tG 0 t ´1 ď G. Then the element tg i P PSL 2 pRq induces an isometry ϕ i : Σ i Ñ Σ 0 , and also a diffeomorphism r ϕ i : PSL 2 pRq Ñ PSL 2 pRq such that µ 1 ˝r ϕ i " µ 1 ptg i ´q P G `pΣ i q for any µ 1 P G `pΣ 0 q.Note that we have rG : tG 0 t ´1s " rG : g ´1 i G 0 g i s " s for all i, since the surfaces Σ 0 , Σ 1 , . . ., Σ s are pairwise isometric (and therefore have the same genus) and since rG : G 0 s " s.
Now since λ Σ is a left Haar measure, we have λ Σ " λ Σ ptg i ´q " λ Σ ˝r ϕ i for all i.It then follows by Lemma 2.22 that Thus ιpµ ptq , λ Σ q " ιpµ, λ Σ q, as required.
Lemma 5.5.Let t P PSL 2 pRq be an element such that G 0 :" t ´1Gt X G has finite index in G. Let µ P G `pΣq be a non-zero current, and define pµ n q 8 n"0 Ă G `pΣq inductively by µ 0 " µ and µ n " µ ptq n´1 for n ě 1.Then the closure of t ř n i"0 c i µ i | n ě 0, c i P r0, 8qu in G `pΣq contains a non-zero current µ such that µ ptq " µ.
Let g 1 , . . ., g s be a right transversal of G 0 in G with g 1 " 1, and for 1 ď i ď s, let Σ i Ñ Σ be the finite covering map corresponding to the subgroup g ´1 i G 0 g i ď G.Note that since µ nm Ñ µ as m Ñ 8, we also have µ nm ptg i ´q Ñ µptg i ´q in G `pΣ i q, and therefore µ ptq nm Ñ µ ptq as m Ñ 8. We aim to show that we also have µ ptq nm Ñ µ as m Ñ 8; this will imply that ş f dµ " ş f dµ ptq for every continuous function f : PSL 2 pRq Ñ R with compact support, and the result will then follow by Theorem 2.14.
Let D be a (relatively compact) fundamental domain for the action of G on PSL 2 pRq by left multiplication, and let K Ă PSL 2 pRq be compact.We claim that µ n pKq ď µpDKq for all n ě 0. Indeed, since we have µ n " µ ptq n´1 " s ´1 ř s i"1 µ n´1 ptg i ´q for all n ě 1 and since µ 0 " µ, it follows by induction on n that µ n " s ´n ř s n i"1 µph i ´q for some h 1 , . . ., h s n P PSL 2 pRq.We can pick some k 1 , . . ., k s n P G such that k i h i P D for each i.Note that µpk i h i ´q " µph i ´q since µ is G-invariant; therefore, Proposition 5.6.Let t P PSL 2 pRq be an elliptic isometry of RH 2 such that t ´1Gt X G has finite index in G and such that xG, ty is dense in PSL 2 pRq.Let F : G `pΣq Ñ r0, 8q be a continuous positively linear function such that F pγ ptq q " F pγq for all γ P C `pΣq.Then F " k ¨ιp´, λ Σ q for some k ě 0.
Proof.Let k " F pλ Σ q{ιpλ Σ , λ Σ q.We will aim to show that F pγq " k¨ιpγ, λ Σ q for all γ P C `pΣq.As R `C`p Σq is dense in G `pΣq [Bon88, Proposition 2] and as F and ιp´, λ Σ q are positively linear and continuous, this will imply the result.

Lattices in the hyperbolic plane and a tree
In this section we will collect a number of results about irreducible cocompact lattices in the product of PSL 2 pRq and the automorphism group T of a locally-finite unimodular leafless tree T .Will assume T is non-discrete.Throughout Γ will be an irreducible cocompact lattice in PSL 2 pRq ˆT .Note that by [Hug21b, Corollary 3.6] Γ is either an irreducible S-arithmetic lattice and T is a pp `1q-regular tree for some prime p, or Γ is non-residually finite.In either case, by Theorem 2.7 Γ contains a commensurated subgroup G isomorphic to the fundamental group of a closed compact surface which arises as a finite index subgroup of a vertex stabiliser of the action of Γ on T .
First, we will investigate the density of the projection of Γ to PSL 2 pRq.
Lemma 6.1.The projection P of Γ to PSL 2 pRq is dense.
Proof.If Γ is linear then P contains an S-arithmetic lattice and such a subgroup of PSL 2 pRq is dense.Thus, we may assume Γ is non-residually finite.By Theorem 2.7 Γ splits as a graph of groups in which each vertex group is a finite extension of a uniform lattice in PSL 2 pRq.In particular, P contains a uniform PSL 2 pRq-lattice and hence is Zariski-dense in PSL 2 pRq.
A Zariski-dense subgroup of SL 2 pRq is either dense or discrete.Indeed the Lie algebra of its closure is an ideal, hence either 0 or sl 2 pRq.Now, since Γ is irreducible, P is non-discrete and so we conclude that P is dense in PSL 2 pRq.
Our next task is to show there is a commensurated surface subgroup of Γ which is M-quasiconvex with respect to any biautomatic structure pB, Mq.
The key fact is that in a biautomatic group the centraliser of a finite set is M-quasiconvex (see Theorem 2.11(i)).Before this we will need a lemma.Lemma 6.2.If Γ is non-residually finite, then we have a short exact sequence where F is fundamental group of a graph of finite groups and P is linear.
In particular, if Γ is torsion-free, then F is a free group.In both cases F is infinite, not virtually abelian, and every locally finite subgroup of F is finite.
Proof.Since Γ is non-residually finite Γ does not admit any faithful linear representation and so F is non-trivial.Now, Γ splits as a graph of finite-by-Fuchsian groups and each Fuchsian group is isomorphic to its image in P .It follows that the action of F on T has finite stabilisers.In particular, F is the fundamental group of a graph of finite groups.If Γ is torsion-free, then each vertex and edge stabiliser of the F -action on T is trivial.It follows that F admits a free action on a tree and so must be free.That P is linear follows from the fact PSL 2 pRq is linear.Since Γ is CATp0q, it has only finitely many conjugacy classes of finite subgroups, implying that any ascending sequence of finite subgroups of Γ terminates.It follows that Γ (and so F ) has no infinite locally finite subgroups.We claim that if F was finite then F must act trivially on T .Indeed, if F was finite then it acts on T elliptically with fixed point set T F a subtree of T .By normality of F in Γ, the subtree T F is Γ-invariant.But Γ is a uniform lattice and T is leafless, so Γ acts minimally on T .Thus, F is infinite.It remains to show that F is not virtually abelian.
Since F is infinite and not locally finite, it contains a finitely generated infinite subgroup.Such a subgroup cannot be torsion (otherwise it would fix a point in T , contradicting the fact that the action of F on T has finite stabilisers); it follows that F contains an infinite order element g.Since the action of F on T has finite stabilisers, g must be hyperbolic in this action; let Ď T be the axis of g.Since T is non-discrete it follows that T is not a line; moreover, since the Γ-action on T is cocompact and since T is leafless and locally-finite, there exists an element h P Γ such that h ‰ .Then g and hgh ´1 are two hyperbolic elements of F that have distinct axes, so g n does not commute with hg m h ´1 for any n, m ‰ 0. This implies that F is not virtually abelian.Proposition 6.3.Suppose pB, Mq is a finite-to-one biautomatic structure on Γ.If Γ is non-residually finite and torsion-free, then any vertex stabiliser of the action on T is an M-quasiconvex subgroup.
Proof.Let G be a commensurated surface subgroup of Γ.Let F " Kerpπ PSL 2 pRq q and note by Lemma 6.2 that F is a non-abelian free subgroup acting freely on T .Let g, h be contained in this free group and suppose that they do not commute.
We claim since G is commensurated and F is normal, the elements g and h commute with the subgroup S " G X G g X G h which has finite index in G. Indeed, let s P S and note s and s g fix vertices of T .It follows that the commutator rs, gs lies in G g .The commutator rs, gs maps trivially under the projection to PSL 2 pRq, but the projection restricted to G g is injective.Thus, rs, gs " 1 and the claim follows.
By the previous claim, C :" C Γ ptg, huq contains S. Now, g and h have distinct axes so C must fix a vertex on T .In particular, C is a finiteindex subgroup of a vertex stabiliser containing a finite index subgroup S of G, implying that C is commensurable with G in Γ.Finally, since C is the centraliser of a finite set if follows from Theorem 2.11 that C is Mquasiconvex.By Lemma 2.12, it follows that G is M-quasiconvex as well.
Finally, we record this proposition for later use.It is a special case of [Hug22b, Corollary 3.3].Proposition 6.4.Γ is a hierarchically hyperbolic group.

An explicit example
Throughout this section we will use quaternion algebras and arithmetic Fuchsian groups derived from them, for the relevant background the reader should consult [Kat92, Chapter 5].The construction appeared in the first author's PhD thesis, however, the example there is different to the one given here [Hug21a, Section 4.5.2].
Let Q be the quaternion algebra p2, 13q Q , this is a 4-dimensional algebra over Q with basis t1, i, j, ku satisfying the relations i 2 " 2, j 2 " 13 and k " ij " ´ji.The algebra Q has a representation ϕ : Q Ñ M 2 pRq given by Conjugating M by t we obtain another maximal order N .Let U 1 pM q and U 1 pN q denote the groups of norm one quaternions under multiplication in M and N respectively.Note that their image under ϕ is contained in SL 2 pRq.Denote the image of U 1 pM q and U 1 pN q under ϕ after projecting to PSL 2 pRq by PM and PN respectively.Both of these groups are isomorphic to the fundamental group of a genus 2 surface (this may be verified in Magma).It is easy to see ϕptq commensurates U 1 pM q and hence U 1 pM q and U 1 pN q share a common finite index subgroup.The intersection K " PM X PN has index 12 in both PM and PN , in particular, K is the fundamental group of a genus 13 surface.We compute, using Dehn's algorithm a word in a, b, c, d for t ´1gt for each generator g of K. We will denote the subgroup generated by these words H and note that H t " K.
We now build a HNN-extension Γ " PM ˚Ht "K .The group has 5 generators which (abusing notation) we label a, b, c, d, t and admits a presentation with 27 relations, displayed in Appendix A.
Proof.Since Γ is a graph of groups equipped with a morphism to PSL 2 pRq such that the vertex stabiliser Γ v is a uniform PSL 2 pRq lattice and the stable letter commensurates Γ v , it follows Γ is a graph of lattices in the sense of Definition 2.6.The two embeddings of the edge group have index 12 in Γ v so the Bass-Serre tree of Γ is 24-regular.Thus, Γ is a uniform lattice in PSL 2 pRq ˆT24 by Theorem 2.7.The image of the subgroup generated by stable letter t in PSL 2 pRq is clearly non-discrete because it is generated by an infinite order elliptic isometry.The irreducibility now follows from [Hug21b, Proposition 3.4].
Proof.Because Γ is an HNN-extension the first Betti number of Γ is at least 1 (in fact a direct computation yields it is exactly 1).Since Γ is an irreducible lattice it follows from [Hug21b, Proposition 3.7] that Γ is non-residually finite.
Lemma 7.3.The translation lengths of a and c in their action on RH 2 are not rational multiples of each other.
The left hand side is always of the form m 1 `m2 ? 5 and the right hand side is always of the form m 3 `m4 ?21 for some rational numbers m 1 , m 2 , m 3 , m 4 ą 0. This is clearly impossible and we conclude that τ paq is not a rational multiple of τ pcq.

Proof of Theorem A
Theorem A. There exists a non-residually finite torsion-free irreducible uniform lattice Γ ă PSL 2 pRq ˆT24 such that Γ is a hierarchically hyperbolic group but is not biautomatic.
Proof.Let Γ be the HNN-extension constructed in Section 7.Then, Γ is an irreducible uniform lattice in PSL 2 pRq ˆT24 by Lemma 7.1, non-residually finite by Lemma 7.2, torsion-free by construction, and a hierarchically hyperbolic group by Proposition 6.4.It remains to show Γ is not biautomatic.
Let p G ă Γ be a vertex stabiliser for the Γ-action on the Bass-Serre tree T 24 of Γ.By construction, we have Γ " x p G, p ty for an element p t P Γ such that t :" π PSL 2 pRq p p tq is an infinite order elliptic isometry of RH 2 .Moreover, the group G :" π PSL 2 pRq p p Gq is a torsion-free uniform lattice in PSL 2 pRq, and the projection π PSL 2 pRq pΓq " xG, ty is dense in PSL 2 pRq by Lemma 6.1.As p G is commensurated in Γ, it follows that t ´1Gt X G has finite index in G. Let Σ " GzRH 2 , so that Σ is a closed orientable hyperbolic surface and Gπ 1 pΣq.Now suppose for contradiction that pB, Mq is a (uniformly finite-to-one) biautomatic structure on Γ.By Proposition 6.3, the subgroup p G ă Γ is M-quasiconvex; let pA, Lq be the biautomatic structure on p G associated to pB, Mq, as given by Theorem 2.11.As π PSL 2 pRq maps p G isomorphically to G, we will identify pA, Lq with a biautomatic structure on G. Consider the function }´} L : G Ñ R, as defined in Definition 3.2.By construction, }´} L is invariant under conjugacy in G, and therefore factors through a Theorem 2.11 (S.Gersten and H. Short; D. B. A. Epstein et al.).Let pB, Mq be a biautomatic structure on a group G.

Figure 3 .
Figure 3.The proof of Lemma 4.1.The blue paths have length ď ν, and the green dashed lines have length ď δ `2β `1.

Figure 4 .
Figure 4.The proof of Proposition 4.2.The top pictures represent the situation in V , the bottom ones in Σ.The red pointis the one at which the quasi-smoothing procedure is done, and the blue paths have length ď λε.
HHP20, Corollary H].It is currently unknown if all coarsely Helly groups are biautomatic, or even if they all are Helly.It has been communicated to us by Alexander Engel and Damian Osajda that they have shown certain mapping class groups are not Helly.Such groups are HHGs and therefore coarsely injective.It is a well known open problem whether S-arithmetic lattices are biautomatic.Indeed, this is a special case of [McC07, Problem 34] in McCammonds list (after the American Institute of Mathematics meeting 'Problems in Geometric Group Theory' April 23-27, 2007 for all g, h P G, we have |gh| L ď |g| L `|h| L `ξ; (ii) for all g, h P G, we have |gh| L ď |g| L `ξ|h| A and |hg| L ď |g| L `ξ|h| A ; (iii) for all g, h P G and w P L such that w represents gh and a prefix of w represents g, we have |g| L `|h| L ď |gh| L `ξ.Proof.Let ν be the constant given by Theorem 2.9.Since CaypG, Aq is hyperbolic, there exists a constant δ such that geodesic triangles in CaypG, Aq are δ-slim, and a constant β such that any two pν, νq-quasi-geodesics with the same endpoints are Hausdorff distance ď β away from each other [CDP90, Chapitre 3, Théorème 1.2].In particular, pν, νq-quasi-geodesic triangles in CaypG, Aq are pδ `2βq-slim.We set ξ :" νp2δ `4β `4ν `3q.(i)Let v 1 ,v 2 , w P L be words representing g, h and gh, respectively, such that |g| L " |v 1 |, |h| L " |v 2 | and |gh| L " |w|.Let γ, ζ 1 , ζ 2 Ď CaypG, Aq be the paths from 1 G to gh (respectively from 1 G to g, from g to gh) labelled by w (respectively v 1 , v 2 ).Then these three paths form a pν, νq-quasi-geodesic triangle in CaypG, Aq, which must be pδ `2βq-slim.Thus, if we write γ " γ 1 γ 2 and ζ 1 " ζ 11 ζ 12 in such a way that the endpoints of γ 1 and ζ 11 are distance ď δ `2β apart and γ 1 is as long as possible, then we can write ζ 2 " ζ 21 ζ 22 in such a way that the starting points of γ 2 and ζ 22 are distance ď δ `2β `1 apart.
that v 1 and v 2 represent g and h, respectively.Since v 1 and v 2 are a prefix and a suffix, respectively, of a word in L, it follows that v 1 u 1 , u 2 v 2 P L for some u 1 , u 2 P A ˚with |u 1 |, |u 2 | ď ν; see Figure 3b.It then follows that |g| L ď |v 1 u 1 | `ν|u 1 | and |h| L ď |u 2 v 2 | `ν|u 2 |.Moreover, we have ˇˇ|gh| L ´|w| ˇˇď ν, implying that n pKq " s : PSL 2 pRq Ñ R be a continuous function with compact support K. Since µ ptq n ´µn " n ´1pµ n ´µq for any n ě 1, we haveˇˇˇż f dµ ptq n ´ż f dµ n ˇˇˇ" n ´1 ˇˇˇż f dµ n ´ż f dµ ˇˇˇď n ´1 ˆˇˇˇż f dµ n ˇˇˇ`ˇˇˇż f dµ ˇˇˇď n ´1 }f} 8 pµ n pKq `µpKqq ď n ´1 }f } 8 pµpDKq `µpKqq , and therefore ˇˇş f dµ ptq n ´ş f dµ n ˇˇÑ 0 as n Ñ 8. On the other hand, since µ nm Ñ µ we have ˇˇş f dµ nm ´ş f dµ ˇˇÑ 0 as m Ñ 8. Since n m Ñ 8 as m Ñ 8, it follows that ˇˇş f dµ ptq nm ´ş f dµ ˇˇÑ 0 as m Ñ 8.But as f was arbitrary, it follows that indeed µ it follows that the image of ϕptq in PSL 2 pRq is an infinite order elliptic isometry of RH 2 .A basis for a maximal order M of Q is given by the following quaternions ta, b, c, du :"