Dead ends and rationality of complete growth series

The complete growth series of a finitely generated group is given by $\sum_{n\ge 0} A_ns^n$, where $A_n$ is the sum of elements of length $n$ in the group semiring. We study the $\mathbb NG$-rationality and $\mathbb NG$-algebraicity of such series. We show that having dead ends of arbitrarily large depths is an obstruction to $\mathbb NG$-rationality. In the case of the $3$-dimensional Heisenberg group $H_3(\mathbb Z)$, we prove that the complete series is not $\mathbb NG$-algebraic for any generating set. Dead ends are also used to show that complete growth series of higher Heisenberg groups are not $\mathbb NG$-rational for specific generating sets. Using a more general version of this obstruction, we prove that complete growth series of some lamplighter groups are not $\mathbb NG$-rational either.

The idea of geometric group theory is to see a group G as a geometric space: fix a finite symmetric generating set S, and endow G with the word metric.Once this is done, we may be interested in the growth of the group, specifically the sequence (a n ) n≥0 counting elements g ∈ G at distance ∥g∥ S = n of the identity.This sequence can be looked at under a "coarse" lens, and turned into a quasi-isometry invariant.In another direction, we may study finer properties of the sequence.This is usually done through the "numerical" growth series of G In this paper, we will be interested in an even richer sequence, namely A n = ∥g∥=n g, seen as elements of the group semiring NG.Just as in the standard case, the sequence (A n ) n gives rise to a growth series: the complete growth series of G G(s) Note that both constructions are strongly related.Indeed, there exists an augmentation map ϵ : NG → N, defined as the morphism sending all group elements to 1.This morphism naturally extends into a morphism ϵ : NG A series with coefficients in a semiring R is R-rational if it satisfies a certain system of linear equations (see §1 for a proper definition).Similarly, the series is R-algebraic if it satisfies a certain system of polynomial equations.
The interest of rationality for numerical growth series is that their coefficients are relatively easy to compute: they satisfy a linear recurrence equation.Moreover precise asymptotics are known.Similarly, knowing that the complete growth series of a group is ZG-rational says that the list of elements of length n satisfies some linear recurrence relations.In some sense, being NG-rational is even computationally better.(See Remark 1.10 for a discussion.)To the best of our knowledge, there were no counterexample that could show that the implications induced by ϵ are not equivalences.In other words, it wasn't clear if being N-rational (resp.N-algebraic) could imply being also NG-rational (resp.NG-algebraic).In the present paper, we prove this is not the case, by providing the examples of Table 1.

N-rational
N-algebraic NG-rational NG-algebraic Virtually abelian Table 1: Summary of our results (in red) together with some relevant known results (in black).The first three lines hold for any finite generating set.
It should be noted our methods are not able to distinguish between Z-rational and ZGrational groups.(Using other methods, the first author was able to do so for a natural submonoid of Thompson's group F .)That being said, there are no known example of groups with Z-rational but non-N-rational numerical growth series, or similarly with ZG-rational but non-NG-rational complete growth series.Therefore, we believe Table 1 should hold with N replaced by Z.
Let us summarize how those results are obtained: The obstruction to NG-rationality relies on existence of dead ends of arbitrarily large depths (aka deep pockets) -or related "saddle elements" -in the group (G, S) of interest.Informally, a dead end of depth D is an element g ∈ G so that no longer element can be found in a ball of radius D around g.The existence of groups with dead ends of arbitrarily large depth is not trivial, and was open for several years, until the first examples were given by [7].Here is a version of this obstruction: Theorem 1 (Corollary 2.7).Let G be an infinite group and S a finite generating set.If (G, S) has dead ends of arbitrarily large depth, then its complete growth series is not NG-rational.
This will apply to a large array of groups, in particular to the discrete Heisenberg group The growth of H 3 (Z) has been under a lot of attention, culminating in results of Duchin and Shapiro [10] showing that its numerical growth series is rational (indeed N-rational) w.r.t.any symmetric generating set.On the other hand, deep pockets were already exhibited by Warshall [24].This already implies that complete growth series for H 3 (Z) is not NG-rational for all generating set.We extend Warshall's results showing that all large commutators are almost dead ends of large depths, more precisely Theorem 2 (Theorem 3.13(a)).Fix S a finite symmetric generating set for H 3 (Z).There exist constants C, M such that, for all D ≫ 1 and n With additional arguments from Carnot-Caratheodory geometry we prove Theorem 3 (Theorem 5.3).The complete growth series of the Heisenberg group H 3 (Z) is not NG-algebraic for all finite symmetric generating set.
In the case of higher Heisenberg groups H 2n+1 (Z), Stoll [22] has shown that the numerical growth series of H 5 (Z) w.r.t. the standard generating set is transcendental (i.e., not Z-algebraic), while for all n ≥ 1 the growth series of H 2n+1 (Z) is N-rational w.r.t.better-behaved "cubical" generating sets.On the other side, we establish a version of Theorem 2 for dead ends in H 2n+1 (Z) w.r.t."cubical-like" generating sets (Theorem 3.13(b); see §3.4 for a definition of "cubical-like").As a direct corollary the complete growth series of H 2n+1 (Z) w.r.t.any cubical-like generating set is not NG-rational.Work in progress of the second author shows that complete growth series of 2-step nilpotent groups are never NG-rational, unless the group is virtually abelian.
Other groups of interest are wreath products L ≀ Q.We will only be interested in generating set of the form S L ⊔ S Q .Deep pockets can be found in any group L ≀ Z where L itself has deep pockets (eg.L = C 2 ) [7].This result was recently extended to wreath products L ≀ Q with Q abelian and any S Q , and even any group and at least one generating set S Q [21].However, as soon as the lamp group L doesn't have deep pockets (eg.L = Z), or for specific (non-abelian) base group Q with specific S Q (given by [21]), the group L ≀ Q doesn't have deep pockets.This led us to introduce "saddle elements" (see §2 for a definition), which then allowed us to extend our techniques to some groups without dead ends.In particular Theorem 4 (Theorem 4.3).Let L be a non-trivial group and S L a symmetric generating set.The complete growth series of the wreath product G = L ≀ Z associated to the standard generating set S L ⊔ {t ± } is not NG-rational.This is to be compared with Theorem (Bartholdi [2,Theorem C.3]).Let L be a group and S L a symmetric generating set.Suppose that the complete growth series of the pair (L, S L ) is NG-algebraic.Then the complete growth series of the wreath products G = L ≀ Z with respect to S L ⊔ {t ± } is NG-algebraic.
For instance both results hold for C 2 ≀ Z.(In this case NG-algebraicity also follows from the unambiguous context-free language of geodesic representatives constructed in [6].) These results extend when replacing Z by "tree-like" groups, i.e., groups of the form C * p 2 * Z * q whose Cayley graphs are m-regular trees (m = p + 2q).Note that, whenever m ≥ 3, Parry [17] proved moreover that the numerical growth series is not rational (indeed not Z-rational).

Rationality and Algebraicity
In this section, we recall the definition of R-rational and R-algebraic formal series, where R is a fixed semiring.In what follows, we will consider R = NG, other relevant cases being R = N, Z and ZG.We then proceed with alternate characterizations.See [8,19] for a complete treatment.Definition 1.1 (Proper systems).An algebraic system over R[s] is a system of equations where P i ∈ R[s] ⟨X 1 , X 2 , . . ., X n ⟩ are polynomials with (a priori) non-commutative coefficients and variables, i.e., finite sums of monomials r 0 X i 1 r 1 X i 2 r 2 . . .X i d r d • s k with d, k ≥ 0 integers, 1 ≤ i j ≤ n and r j ∈ R. A system is proper if it contains • no constant term (i.e., no monomial with d = k = 0); • no monomial with d = 1 and k = 0.
A system is linear if furthermore all monomials satisfy d = 0, or d = 1 and The main motivation for those two definitions is the following result: Finally, we say a proper series is R-rational (resp.R-algebraic) if it is solution to a proper linear (resp.algebraic system).More generally Definition 1.4 (R-rational and R-algebraic series).A formal series to a proper linear (resp.algebraic) system over R[s].
Let us give another less technical definition of rational series over non-commutative semirings.We first need the notion of quasi-inverse: Definition 1.5.Given a proper series S(s) ∈ R[[s]], we define its quasi-inverse as S(s) * = n≥0 S(s) n , where S(s) n is the multiplication of S(s) with itself n times and S(s) 0 = 1.
The name quasi-inverse comes from the "equality" (1 − S)S * = 1.This (linear) equation shows that, if S itself is rational, then S * is rational too.Reciprocally, all rational series can be obtained from polynomials using addition, multiplication and quasi-inversion: Rational and algebraic series can also be characterized in terms of automata and languages.
] is recognized by the automaton A if there exist two initial and terminal vertices I, T ∈ V such that where P (I, T ) is the set of all oriented paths from I to T , and p(e 1 . . .e ℓ ) := p(e 1 ) . . .p(e ℓ ).
(a) In case R = NG, we may and will assume that all labels are of the form p(e) = w(e) • s l(e) for some w(e) ∈ G and l(e) ∈ N, at the cost of allowing multiple edges between vertices.
(b) If furthermore S(s) is the complete growth series, then ∥w(γ)∥ = l(γ) holds for all path γ ∈ P (I, T ) (with the obvious notations) Remark 1.10.The computational advantages of NG-rational series over ZG-rational series are made clearer through the formalism of R[s]-automata.Given an NG[s]-automaton for a complete growth series, we can pick a path at random, compute the associated term g • s l , and get a certificate that the element g has length l.In contrast, if the automaton had labels in ZG[s], the term g • s l might cancel out with other terms, so that we need to compute all terms with the same exponent s l before having a certificate g has indeed length l.
Theorem 1.8 admits a generalization linking R-algebraic series and weighted pushdown automata (see [8,Section 7.5.3]).For complete growth series, this can be reformulated as Theorem 1.11.Let G be a group and S a generating set.The associated complete growth series is NG-algebraic if and only if there exists an unambiguous context-free language L ⊂ (G × N) * such that the evaluation map ev : L → G × N is injective and its image is given by ev(L) = {(g, ∥g∥ S ) : g ∈ G}.

Dead ends and saddle elements
Let us recall the notion of dead ends introduced in [3], and introduce some related notions: Definition 2.1.Consider (G, ∥ • ∥) a group endowed with a left-invariant metric (e.g., a finitely generated group endowed with a word metric, or a nilpotent Lie group endowed with a Carnot-Cartheodory metric) and let D > 0 and M ≥ 0 be real numbers.An element g ∈ G is We say that (G, ∥ • ∥) has deep pockets if it possesses dead ends of arbitrarily large depths.
Remark 2.2.Finite (and more generally compact) groups have deep pockets: element of maximal length are dead ends of infinite depth.The notion of deep pockets gets way more interesting when looking at non-bounded groups, as all elements have finite depth.
To the best of our knowledge, the notion of almost dead ends is new, even though almost dead ends appear implicitly in Warshall's work [25,24].The main interest of almost dead ends is that the existence of almost dead ends of large depth is preserved by rough isometries.Definition 2.3.Let X, Y be two metric spaces.A function f : X → Y is a rough isometry (or (1, K)-quasi-isometry) if there exists some constant K ≥ 0 such that • for all x, x ′ ∈ X, we have • for all y ∈ Y , there exists x ∈ X such that d Y y, f (x) ≤ K.
Remark.Even though this might not be obvious from the definition, this defines an equivalence relation on metric spaces, in particular it is symmetric.and (H, d H ) be two roughly isometric metric group.Suppose that there exists some M ≥ 0 such that G contains almost dead ends of arbitrarily large depth and fixed margin M , then the same is true in H.
Proof.Let f : G → H be a rough isometry.Up to translation we may suppose that f (e G ) = e H . Let g ∈ G be an almost dead end of margin M and depth D. We prove that f (g) is an almost dead end.For any h By hypothesis g is an almost dead end of depth D so that ∥x∥ ≤ ∥g∥ + M hence This means that f (g) is an almost dead end of margin M + 3K (which is fixed) and depth D − 2K (which can be made arbitrarily large for well chosen g ∈ G).
Usually we should think of the situation where the depth D is way larger than the margin M .In case of integer-valued metric and D ≫ M , the following key lemma ensures that we can "promote" almost dead ends into genuine dead ends of comparatively large depth.Lemma 2.5 (Warshall [25,Proposition 6]).Let X be a metric space, f : X → N a function, and D, M ∈ N. Suppose that there exists x ∈ X such that Then there exists Let us now relate dead ends with the rationality of complete growth series Theorem 2.6 (Pumping lemma + ε).Let G be a group and S a finite generating set.Suppose that the complete growth series of (G, S) is NG-rational.Then there exists D > 0 such that all but finitely many g ∈ G can be written as g = uv with v ̸ = e, ∥v∥ ≤ D and is the translation number of v.
Proof.As the complete growth series is NG-rational, it is recognized by an automaton (V, E).
We take D = 2 |V | max e∈E ∥w(e)∥ (with p(e) = w(e)s l(e) as in Remark 1.9).Note that each element g ∈ G corresponds to a path γ g ∈ P (I, T ).Moreover, all but finitely many g correspond to paths containing cycles.In that case, we decompose γ g as a concatenation αβδ with β a non-empty simple cycle and δ a simple path.Among the |V | + 1 last visited vertices, at least one is repeated.The loop between the two last visits of this vertex is β, and everything coming afterward is δ.In particular the path βδ is formed of at most |V | edges.
Take u = w(αδ), v = w(δ) −1 w(β)w(δ) and τ = l(β) ∈ N. Note that ∥v∥ ≤ D as v is a product of at most 2 |V | edge labels w(e).Moreover αβ n δ ∈ P (I, T ) for all n, so that we get terms w(αβ n δ) • s l(αβ n δ) in the complete growth series.It follows that Finally, let us look at τ .The previous equality implies that τ ̸ = 0, otherwise we would get infinitely many elements inside the sphere of radius ∥u∥.Moreover the triangle inequality gives n is indeed the translation number of v.
Theorem 1 announced in the introduction follows as a quick corollary: (ii) (G, S) contains an infinite sequence of distinct saddle elements of increasing depth, then its complete growth series of (G, S) is not NG-rational.
Proof.We argue by contraposition.Suppose that (G, S) is infinite and its complete growth series is NG-rational.By Theorem 2.6 all but finitely many elements can be written as g = uv with ∥v∥ ≤ D and ∥uv n ∥ = ∥u∥ + nτ for all n ≥ 0. It follows that (i) In particular g cannot be a dead end of depth ≥ D. As G is infinite, all elements have finite depth, in particular the finitely many elements left out by this argument have finite depth.
It follows that (G, S) doesn't have deep pockets.
(ii) In particular g cannot be a saddle element of depth ≥ D. As only finitely many elements are left out, G cannot contain an infinite sequence with the required condition.

Dead ends in Heisenberg groups
We reprove one of Warshall's results, in order to retrieve the finer conclusions of Theorem 2, which will then allow us to find dead ends in higher Heisenberg groups.
3.1 Models for H 3 (R) and H 3 (Z) Elements of H 3 (R) can be seen as equivalence classes of absolutely continuous curves in R 2 starting at 0, with concatenation as operation (see Figure 2).We associate to each curve γ Proposition 3.1.Given two paths g, h, their concatenation gh has parameters gh = ĝ + ĥ, det(ĝ, ĥ).
In particular, the operation "concatenation" is well-defined on H 3 (R).With the constant curve as neutral element, and curves traveled backward as inverses, this defines a group.An alternate formula for A(γ) helps to better understand this result.We define a closed curve γ c by concatenating γ with the segment from γ to 0, and a function w γ : R 2 \ Im(γ c ) → Z defined as w γ (x, y) = winding number of γ c around (x, y) (see Figure 2).Then 1 "Balayer" is french for "to sweep".A(γ) is the area swept by the moving segment from 0 to γ(t).
The discrete Heisenberg group H 3 (Z) can be seen as the subgroup generated by the unit segments x, y from the origin 0 to (1, 0), (0, 1) respectively.Moreover, we denote z = [x, y].

CC-metrics on H 3 (R)
We recall the definition of the Carnot-Caratheodory metric on H 3 (R).First Definition 3.2 (Minkowski norm).Let L ⊂ R n be a centrally symmetric (full dimensional) convex polytope.We define the Minkowski norm on R n as ∥v∥ L = min{λ ≥ 0 : v ∈ λL}.
This norm is particularly nice among subFinsler metrics as it is homogeneous: Proposition 3.4 (Dilation).Given λ ∈ R, we define the dilation δ λ : H 3 (R) → H 3 (R) as the map sending each path to the image of this path by an homothety of ratio λ (centered at 0).The map δ λ is an automorphism (for λ ̸ = 0) satisfying CC-metrics on H 3 (R) can be described precisely.First a classical result of Busemann on the isoperimetric problem in the plane with Minkowski metric: Theorem 3.5 (Busemann [4]).Fix L ⊂ R 2 centrally symmetric convex polygon.Among closed curves, the maximal ratio of enclosed area to the square of the perimeter is achieved (uniquely up to scaling) for the isoperimetrix I, i.e., the boundary of the rotation by ± π 2 of the polar dual In other words, for any closed curve γ, we have  Note that, whenever L is polygonal, then I is polygonal with sides parallels to vertices of L.
This was generalized for all CC-geodesics in a beautiful paper of Duchin and Mooney: • Regular geodesics which follows a portion of a dilate of the isoperimetrix.
• Unstable geodesics for which all tangent directions γ ′ (t) lie in a common positive cone spanned by two consecutive vertices of L. These coincides with geodesics in (R 2 , ∥ • ∥ L ).
Moreover all such paths are CC-geodesics.

Almost dead ends in H 3 (R) for CC-metrics
It should first be noted that H = H 3 (R) doesn't have any dead end 2 .Indeed, it can easily be seen from the Duchin-Mooney Structure Theorem that CC-geodesics for elements g / ∈ [H, H] can be extended into geodesics for longer elements.Moreover, if g = z A with A ̸ = 0.For any D > 0, we can take h = z ±ε which has CC-length √ ϵ ∥z∥ CC ≤ D. We can chose the sign such that the equality |A ± ε| > |A| is true.It follows that for the right choice of sign.This means we can only hope for almost dead ends in (H 3 (R), ∥ • ∥ CC ), which happens to be sufficient to draw conclusions in the word metric case.Proof.Fix g, h ∈ H 3 (R) as in the statement.There exist consecutive vertices ŝi , ŝi+1 of L such that ĥ lives in the positive cone spanned by ŝi , ŝi+1 , say ĥ = αŝ i + βŝ i+1 with 0 ≤ α, β ≤ D. We define B( ĥ) as the area of the triangle with sides ĥ, −αŝ i and −βŝ i+1 (in that order).ĥ αŝ i βŝ i+1 B( ĥ) Note that the area A(h) + B( ĥ) is enclosed by a curve of length ∥h∥ CC + ∥ ĥ∥ L ≤ 2D (specifically h concatenated with the two segments −αŝ i and −βŝ i+1 ), so that 2 There is another definition of dead ends in pointed geodesic metric spaces.A dead end is an element g such that no geodesic leading to g can be extended into a geodesic for a longer element.With this definition, non-trivial commutators in H3(R) are indeed dead ends.
Consider γ z any geodesic representing z and consider the dilation δ λ γ z with We consider D ≫ 1 so that all sides of δ λ γ z have Minkowski-length ≥ D. In particular we can find two points h 0 and h 1 on the curve δ λ γ z , more precisely on the sides with direction −ŝ i+1 and −ŝ i respectively, differing by a vector ĥ.Up to picking a different starting point on the geodesic δ λ γ z , we may assume h 0 = 0 and h 1 = ĥ.
It remains to bound the length of β to estimate ∥gh∥ CC .We have

CC
and therefore

Almost dead ends in higher Heisenberg groups
We first recall a construction from [16], using some formalism from [22]: Definition 3.9 (Centrally amalgamated direct products, see [22]).Let (G i ) i=1,...,m , Z be groups, and Note that each G i can be naturally identified with a subgroup of Z G i .
To grasp some intuition, let us talk about H 2m+1 (R) = Z G i , where G i ≃ H 3 (R), and Z = {z n | n ∈ R} is the common center.Once again, elements of H 2m+1 (R) can be seen as equivalence classes of curves, more specifically curves in (R 2 ) m starting at 0. A curve projects on each of the m planes, where we can compute an area.Centrally amalgamating means we add those m areas together, two curves are equivalent if they have same endpoints and sum of areas.This point of view allows to define CC-metrics on H 2m+1 (R) for any Minkowski norm on R 2m .Definition 3.10.Given metrics ∥ • ∥ i on each G i , we can endow Z G i with two natural metrics: • The cubical-like metric If we start with word metrics ∥ • ∥ S i on each G i , both metrics are word metrics corresponding to • the standard-like generating set S stnd−like = i S i .
• the cubical-like generating set S cube−like = i (S i ∪ {e}).
Going back to the H 2m+1 (R) example, if we start with CC-metrics ∥ • ∥ CC,L i on each G i , we get two CC-metrics on H 2m+1 (R) corresponding to respectively.Geodesic curves representing z n can be understood in both cases: • With standard-like metric, any geodesic representing z n projects to a geodesic for z n in one of the planes where ∥z∥ 2 CC,L i is minimal, and to constant curves in all other planes.• With cubical-like metric, any geodesic representing z n projects to geodesics for some central element z n i in each plane, and those geodesics have the same length.
Note that, if the min in (1) (resp.( 2)) is achieved for g i z i 's that are all almost dead ends of depth D and margin M i , then [g 1 , . . ., g n ] is an almost dead end of depth D and margin i M i (resp.max i M i ).This leads to the following generalization of Theorem 3.8: Proposition 3.11.Consider the higher Heisenberg group where G i ≃ H 3 (R), and Z is the common center.Fix polygonal CC-metric ∥ • ∥ CC,i on each G i , then Theorem 2 holds in (H 2m+1 (R), ∥ • ∥ CC,cube−like ), i.e., there exist constants C, M such that, for all D ≫ 1 and Consider g = z n for n ≥ C • D 4 and D ≫ 1.Note that -for g = z n -formula (2) reads as (b) As (G j , S j ) doesn't have any deep pocket, there exists some D such that (G j , S j ) doesn't have any dead end of depth ≥ D. We show that the same holds in ( i G i , S stnd−like ).Consider any g ∈ i G i .As g j ∈ G j isn't a dead end of depth D, there exists h j ∈ G j with ∥h j ∥ S j ≤ D and ∥g j h j ∥ S j > ∥g j ∥ S j .We define h = (e, . . ., e, h j , e, . . ., e) ∈ i G i .We have

Lamplighters and wreath products
Let us start with a classical result for lengths in wreath products: Elements g ∈ L≀Q can be identified with pairs (Φ, q) with Φ : Q → L a finitely supported function, and q ∈ Q.Moreover, the length of g is given by the formula ∥g∥ std = p∈Q ∥Φ(p)∥ S L + T S(e Q ; supp(Φ); q), where T S(x; S; y) is the length of the shortest path in the Cayley graph Cay(Q, S Q ) starting at x, going through S in some order, and ending at y.
Using this formula, it was shown in [7] that many of those groups have deep pockets: Theorem 4.2 (Theorem 6.1 in [7]).Consider G = L ≀ Z with L non-trivial, endowed with the standard generating set S = S L ∪ {t ± }. Suppose that L has dead ends of arbitrary depth w.r.t.S L (for instance if L is finite), then so does G with respect to S.
Note that the condition on (L, S L ) is indeed important, as wreath products like Z ≀ Z do not have dead ends (at least w.r.t. the standard generating set).However, the question of NG-rationality of their complete growth series is still settled by the following proposition: As a corollary, the associated complete growth series in not NG-rational.
Proof.Let ℓ ∈ L \ {e L } and define We consider g d = (Ψ d , 0) i.e., the element with only lamps in a non-trivial state on site ±d, and the lamplighter guy back at 0. Consider h = (Φ, q) ∈ L ≀ Z with ∥h∥ S ≤ d.We have It follows that ∥g d h∥ S can be easily computed: (We use that any path going through ±d must go through the entire interval.)Similarly Compare both formula, recalling that S L is symmetric: we have ∥g d h∥ S = g d h −1 S .Remark 4.4.Both Theorem 4.2 and Proposition 4.3 can be generalized when (Z, {t ± }) is replaced by (Q, S Q ) such that the associated Cayley graph is an infinite tree.(So Q is a free product of copies of Z and C 2 .)The element g d can be taken to be g However, in these cases, [17] already concludes non Z-rationality for numerical growth series.

Context-free languages of polynomial growth and Dyck loops
A k-Dyck word is a word over the set of symbols • each symbol appears exactly once, • [ i appears before ] i , • and if [ j appears in between [ i and ] i , then so does ] j .
For instance, [ and ] 1 are not.
We fix A an alphabet and a k-Dyck word z.A k-Dyck loop with underlying word z is the set of words obtained by placing fixed words w 0 , . . ., w 2k ∈ A * in between the parenthesis, and replacing parenthesis [ i and ] i by powers u n i i and v n i i respectively, with u i , v i ∈ A * fixed, and n i any positive integer.For instance, (a) Suppose on the contrary 0 / ∈ ConvHull{û i + vi }, then there exists a linear form h : R 2 → R such that h(û i + vi ) > 0 for all i.Let m > 0 be the minimum of those k values.We have so that h(ĝ) > 0 except for finitely many choices of n 1 , n 2 , . . ., n k .It follows that ĝ = 0 (i.e., g ∈ [G, G]) for only finitely many values of the parameters, a contradiction.
(b) Consider g 0 = g(n 1 , n 2 , . . ., n k ) any commutator appearing in our Dyck loop.Define Note that all those g n are distinct commutators.Theorem 3.13 tells us that for all D ≫ 1, there is a number N D such that for n ≥ N D , g n is an almost dead end of depth ≥ D and margin M .Now consider for any n Recall that the underlying word starts and ends by [ 1 and ] 1 respectively, and that g n is a commutator (hence is central in G).It follows that we can rewrite w, D = ∥h∥ S and n ≥ N D .We have g n and a nearby element gn = g n h.We compute their lengths in order to find a contradiction.As both g n and gn appears in the Dyck loop, we know We have ∥g n h∥ S > ∥g n ∥ S + M which is a contradiction with g n being an almost dead end of depth D and margin M .
Fix m, n ∈ N. We are going to consider g 1 (mλ 1 , . . ., mλ j−1 ) and g 2 (nλ j , . . ., nλ k ) and approximate them by better behaved δ m h 1 and δ n h 2 .In order not to crumble under notations, let us take a specific underlying word [ For each g ∈ H 3 (R), we define ḡ ∈ H 3 (R) the element represented by a straight line segment from 0 to ĝ.The elements h 1 and h 2 are defined as follows: where the second ∼ follows from Krat's theorem, or even Pansu's theorem.Similarly Let us first consider m = n = 1.We know that h 1 h 2 is a commutator, so any geodesic has to follow a rescaled isoperimetrix.Suppose wlog that ∥h 1 ∥ CC ≥ ∥h 2 ∥ CC , then any geodesic γ 1 for h 1 has to cover at least half the perimeter of the isoperimetrix.
• If either the isoperimetrix has ≥ 6 sides or ∥h 1 ∥ CC > ∥h 2 ∥ CC , then γ 1 has to cover two corners of the isoperimetrix, hence the scale of the isoperimetrix followed by any geodesic continuation of γ 1 is fixed, and γ 1 γ 2 is the maximal geodesic continuation.In particular the longer curve γ 1 • δ 2 γ 2 cannot be a geodesic continuation.• The only remaining case appears when the isoperimetrix has 4 sides, ∥h 1 ∥ CC = ∥h 2 ∥ CC and the geodesics γ 1 and γ 2 meet at corners of the isoperimetrix.However we still have the same contradiction as γ 1 • δ 2 γ 2 is not a CC-geodesic.Remark.This argument extends for H 2n+1 (Z) with cubical-like generating sets.

Figure 1 :
Figure 1: Implications between different properties

Corollary 2 . 7 .
Let G be a group and S a finite generating set.If either (i) G is infinite and (G, S) has deep pockets, or

Figure 2 :
Figure 2: Concatenation of two elements g, h and a few winding numbers.

Figure 3 :
Figure 3: Examples of L, L * and I.

Figure 4 :Remark 3 . 7 .
Figure 4: A few CC-geodesics of both types for the previous choice of L. Remark 3.7.It follows that, for any g ∈ [H, H], all CC-geodesics follow the same re-scaled copy of I. Specifically, if g = z A (that is, A(g) = A), the scaling factor should be |A| / |A(I)|.Only the starting point along the curve is up to choice.In particular,

Proposition 4 . 1 .
Consider G = L ≀ Q, endowed with the standard generating set S = S L ∪ S Q .

Theorem 4 . 3 .
Let G = L ≀ Z with L non-trivial.Consider the standard generating set S = S L ∪ {t ± } with S L symmetric.There exists a sequence of distinct elements (g d ) satisfying ∀h ∈ G such that ∥h∥ S ≤ d, ∥g d h∥ S = ∥g d h −1 ∥ S .

γ 1 Figure 6 :
Figure 6: All geodesic continuations of γ 1 have to follow the dotted path.
[18]ad of depth ≥ D.Remark 3.15.Proposition 3.13 allows to recover results due to Riley and Warshall in[18], namely examples of groups with deep pockets for some generating sets, and without deep pockets for others.Their examples are (C 2 ≀ Z) × Z and Γ 2 (Z/2Z) × Z (here Γ 2 (Z/2Z) is the Baumslag-Remeslennikov group, a finitely presented metabelian group), with what happens to be cubicallike and standard-like generating sets.We propose H 3 (Z) × Z (with cubical-like and standard generating sets) as an arguably simpler finitely presented example.