On the subexponential growth of groups acting on rooted trees

Francoeur, Dominik

doi:10.4171/ggd/531

Cited by 5 publications

(5 citation statements)

References 9 publications

(14 reference statements)

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“…We deduce that none of these groups has bounded generation. Combining this result with known results on intermediate growth among spinal groups (see [15, 18]), we get new examples of groups without rational cross‐sections. Corollary The following spinal groups do not admit any rational cross‐section.…”

Section: Monsters Without Rational Cross‐sectionssupporting

confidence: 55%

“…This includes the (nontorsion) Grigorchuk-Erschler group 𝐺 (01) ∞ . (b) Spinal groups 𝐺 𝜔 with 𝑝 = 3 satisfying hypothesis of [15,Theorem 4.6]. This includes the Fabrykowski-Gupta group.…”

Section: Groups Acting On Regular Rooted Treesmentioning

confidence: 99%

“… (a)Non virtually‐abelian spinal groups

G_\omega

with

p=2

. This includes the (nontorsion) Grigorchuk–Erschler group

G_{(01)^\infty }

. (b)Spinal groups

G_\omega

with

p=3

satisfying hypothesis of [15, Theorem 4.6]. This includes the Fabrykowski–Gupta group. …”

Section: Monsters Without Rational Cross‐sectionsmentioning

confidence: 99%

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Rational cross‐sections, bounded generation, and orders on groups

Bodart

2024

Journal of London Math Soc

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We provide new examples of groups without rational cross‐sections (also called regular normal forms), using connections with bounded generation and rational orders on groups. Our examples contain a finitely presented HNN‐extension of the first Grigorchuk group. This last group is the first example of finitely presented group with solvable word problem and without rational cross‐sections. It is also not autostackable, and has no left‐regular complete rewriting system.

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Section: Monsters Without Rational Cross‐sectionssupporting

confidence: 55%

Section: Groups Acting On Regular Rooted Treesmentioning

confidence: 99%

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Rational cross‐sections, bounded generation, and orders on groups

Bodart

2024

Journal of London Math Soc

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“…First introduced technique for getting an upper bound for G uses the strong contraction property [12] (also known as sum contraction property), which says that there is a finite index subgroup H of G such that any element g ∈ H can be uniquely decomposed into some elements, whose sum of lengths in not larger than C|g| + D, where 0 < C < 1 and D are constants independent of g [12]. Later this technique was developed and many variants were introduced [2,7,9]. In 2004, to get a lower bound for a certain class of groups of intermediate growth, Anna Erschler introduced a method for partial description of the Poisson boundary [7].…”

Section: Introductionmentioning

confidence: 99%

On growth of generalized Grigorchuk's overgroups

Samarakoon

2020

ADM

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Grigorchuk's Overgroup G˜, is a branch group of intermediate growth. It contains the first Grigorchuk's torsion group G of intermediate growth constructed in 1980, but also has elements of infinite order. Its growth is substantially greater than the growth of G. The group G, corresponding to the sequence (012)∞=012012…, is a member of the family {Gω|ω∈Ω={0,1,2}N} consisting of groups of intermediate growth when sequence ω is not eventually constant. Following this construction we define the family {G˜ω,ω∈Ω} of generalized overgroups. Then G˜=G˜(012)∞ and Gω is a subgroup of G˜ω for each ω∈Ω. We prove, if ω is eventually constant, then G˜ω is of polynomial growth and if ω is not eventually constant, then G˜ω is of intermediate growth.

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“…All spinal groups with d = 2 are of intermediate growth. For d ≥ 3, this is known for some but not all of them ( [3,4,14]). All spinal groups are amenable [25].…”

Section: Introductionmentioning

confidence: 99%

On spectra and spectral measures of Schreier and Cayley graphs

Grigorchuk,

Nagnibeda,

Pérez

2020

Preprint

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We are interested in various aspects of spectral rigidity of Cayley and Schreier graphs of finitely generated groups. For each pair of integers d ≥ 2 and m ≥ 1, we consider an uncountable family of groups of automorphisms of the rooted d-regular tree which provide examples of the following interesting phenomena. For d = 2 and any m ≥ 2, we get an uncountable family of non quasi-isometric Cayley graphs with the same Laplacian spectrum, absolutely continuous on the union of two intervals, that we compute explicitly. Some of the groups provide examples where the spectrum of the Cayley graph is connected for one generating set and has a gap for another.For each d ≥ 3, m ≥ 1, we exhibit infinite Schreier graphs of these groups with the spectrum a Cantor set of Lebesgue measure zero union a countable set of isolated points accumulating on it. The Kesten spectral measures of the Laplacian on these Schreier graphs are discrete and concentrated on the isolated points. We construct moreover a complete system of eigenfunctions which are strongly localized.

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On the subexponential growth of groups acting on rooted trees

Abstract: We show that every group in a large family of (not necessarily torsion) spinal groups acting on the ternary rooted tree is of subexponential growth.

Cited by 5 publications

References 9 publications

Rational cross‐sections, bounded generation, and orders on groups

Rational cross‐sections, bounded generation, and orders on groups

On growth of generalized Grigorchuk's overgroups

On spectra and spectral measures of Schreier and Cayley graphs

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