Subgroup Growth of Virtually Cyclic Right-Angled Coxeter Groups and Their Free Products

We show that any graph product of finitely generated groups is hierarchically hyperbolic relative to its vertex groups. We apply this result to answer two questions of Behrstock, Hagen, and Sisto: we show that the syllable metric on any graph product forms a hierarchically hyperbolic space, and that graph products of hierarchically hyperbolic groups are themselves hierarchically hyperbolic groups. This last result is a strengthening of a result of Berlai and Robbio by removing the need for extra hypotheses on the vertex groups. We also answer two questions of Genevois about the geometry of the electrification of a graph product of finite groups.

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“…By definition, V S W and ˇU W coincides with V W , the relative projection of V to W in S . If d U .x; y/ > E 2 1 C E 1 , then we have…”

Section: • Ifmentioning

confidence: 99%

Hierarchical hyperbolicity of graph products

Berlyne

Russell

2022

Groups Geom. Dyn.

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“…This fact motivates the following questions. [1] x 1168 [1,1] x 30 [2] x 467 14 29359 7992 21367 [1] x 11886 [1,1] x 57 [2] x 8161 [3] x 1263 16 396196 66616 329580 [1] x 140507 [1,1] x 2149 [1,1,1] x 4 [1,2] x 247 [2] x 136833 [2,2] x 40 [3] x 49800 Question 6.1. Let P c (n) be the set of pavings with n darts, all of fixed complexity c (although not necessarily of a fixed homeomorphism type).…”

Section: Counting Pavings Of the Three-spherementioning

confidence: 99%

“…General subgroup growth is the subject of the book [20], and further information on subgroup growth in free products of cyclic groups can be found in [3,[26][27][28][29][30]40]. There, the general theory of subgroup structure in free products of (finite and infinite) cyclic groups is enhanced by using the methods of representation theory, analytic number theory and probability theory, among other tools.…”

Section: Introductionmentioning

confidence: 99%

Three-dimensional maps and subgroup growth

2021

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In this paper we derive a generating series for the number of cellular complexes known as pavings or three-dimensional maps, on n darts, thus solving an analogue of Tutte’s problem in dimension three. The generating series we derive also counts free subgroups of index n in $$\Delta ^+ = {\mathbb {Z}}_2*{\mathbb {Z}}_2*{\mathbb {Z}}_2$$ Δ + = Z 2 ∗ Z 2 ∗ Z 2 via a simple bijection between pavings and finite index subgroups which can be deduced from the action of $$\Delta ^+$$ Δ + on the cosets of a given subgroup. We then show that this generating series is non-holonomic. Furthermore, we provide and study the generating series for isomorphism classes of pavings, which correspond to conjugacy classes of free subgroups of finite index in $$\Delta ^+$$ Δ + . Computational experiments performed with software designed by the authors provide some statistics about the topology and combinatorics of pavings on $$n\le 16$$ n ≤ 16 darts.

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“…for some C in the case Γ is virtually abelian (this follows easily from the fact that t n pΓq ď n A n!, and the recursive formula [17,Lemma 1.1.3]). In the companion paper [3] we study the finer growth of h n and s n for some examples of virtually abelian Coxeter groups.…”

Section: Examplesmentioning

confidence: 99%