Exponential growth rates of free and amalgamated products

Bucher, Michelle; Talambutsa, Alexey

doi:10.1007/s11856-016-1299-4

Cited by 7 publications

(8 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Together with the equalities (1.4) proven in Lemma 11 this immediately implies Theorem 1, except in the case of BS (1,2). For this last group, a finer analysis of its action on its Bass-Serre tree will be needed.…”

Section: Introductionmentioning

confidence: 59%

“…In view of Lemma 11, Theorem 1 follows immediately from Theorem 2 except in the case of BS(1, 2) where we need a better understanding of its action on the Bass-Serre to obtain the accurate lower bound of ω (BS(1, 2), {a, t}) = β, where β is the unique real root of x 3 − x 2 − 2. Let S be a generating set for BS (1,2). As in the proof of Theorem 2, the case where S contains two hyperbolic elements with different axis immediately gives the lower bound of ω(BS(1, 2), S) ≥ 2 > β.…”

Section: Proofs Of Theorems 1 Andmentioning

confidence: 93%

“…Except for this example, very few exact values for Ω(G) have been computed. Known cases include free products Z 2 * Z p k [12] (the cases p k = 3, 4 were proven earlier in [8]), the free product Z 2 * (Z 2 × Z 2 ) and the Coxeter group PGL(2, Z) [2] and a few more examples in the references [2,12,8]. But the question of de la Harpe and Grigorchuk whether Ω(π 1 (Σ g )) is realized on the canonical generators of the fundamental group of a closed surface Σ g with g ≥ 2 is still open (see [5, p.55]).…”

Section: Introductionmentioning

confidence: 92%

“…(c) The growth rate ω(BS(1,2), {a, t}) is equal to the positive root ofx 3 − x 2 − 2.Proof. (a) This part was already shown in the Proposition 8.…”

mentioning

confidence: 98%

“…Notably ψ = Ω(GL(2, Z)) = Ω(P GL(2, Z)), see[2] for more information about this number.EXPONENTIAL GROWTH RATES OF BAUMSLAG-SOLITAR GROUPS…”

mentioning

confidence: 99%

See 4 more Smart Citations

Minimal exponential growth rates of metabelian Baumslag–Solitar groups and lamplighter groups

Bucher¹,

Talambutsa²

2017

Groups Geom. Dyn.

Self Cite

View full text Add to dashboard Cite

Abstract. We prove that for any prime p ≥ 3 the minimal exponential growth rate of the Baumslag-Solitar group BS(1, p) and the lamplighter group Lp = (Z/pZ) ≀ Z are equal. We also show that for p = 2 this claim is not true and the growth rate of BS(1, 2) is equal to the positive root of x 3 −x 2 −2, whilst the one of the lamplighter group L 2 is equal to the golden ratio (1 + √ 5)/2. The latter value also serves to show that the lower bound of A.Mann from [8] for the growth rates of non-semidirect HNN extensions is optimal.

show abstract

Section: Introductionmentioning

confidence: 59%

Section: Proofs Of Theorems 1 Andmentioning

confidence: 93%

Section: Introductionmentioning

confidence: 92%

“…(c) The growth rate ω(BS(1,2), {a, t}) is equal to the positive root ofx 3 − x 2 − 2.Proof. (a) This part was already shown in the Proposition 8.…”

mentioning

confidence: 98%

“…Notably ψ = Ω(GL(2, Z)) = Ω(P GL(2, Z)), see[2] for more information about this number.EXPONENTIAL GROWTH RATES OF BAUMSLAG-SOLITAR GROUPS…”

mentioning

confidence: 99%

See 3 more Smart Citations

Minimal exponential growth rates of metabelian Baumslag–Solitar groups and lamplighter groups

Bucher¹,

Talambutsa²

2017

Groups Geom. Dyn.

Self Cite

View full text Add to dashboard Cite

show abstract

Cubic graphs and the golden mean

Grimmett

2020

Discrete Mathematics

View full text Add to dashboard Cite

The connective constant µ(G) of a graph G is the exponential growth rate of the number of self-avoiding walks starting at a given vertex. We investigate the validity of the inequality µ ≥ φ for infinite, transitive, simple, cubic graphs, where φ := 1 2 (1 + √ 5) is the golden mean. The inequality is proved for several families of graphs including (i) Cayley graphs of infinite groups with three generators and strictly positive first Betti number, (ii) infinite, transitive, topologically locally finite (TLF) planar, cubic graphs, and (iii) cubic Cayley graphs with two ends. Bounds for µ are presented for transitive cubic graphs with girth either 3 or 4, and for certain quasi-transitive cubic graphs.

show abstract

Equivariant K-Homology and K-Theory for Some Discrete Planar Affine Groups

Flores,

Pooya,

Valette

2024

International Mathematics Research Notices

View full text Add to dashboard Cite

We consider the semi-direct products $G={\mathbb{Z}}^{2}\rtimes GL_{2}({\mathbb{Z}}), {\mathbb{Z}}^{2}\rtimes SL_{2}({\mathbb{Z}})$, and ${\mathbb{Z}}^{2}\rtimes \Gamma (2)$ (where $\Gamma (2)$ is the congruence subgroup of level 2). For each of them, we compute both sides of the Baum–Connes conjecture, namely the equivariant $K$-homology of the classifying space $\underline{E}G$ for proper actions on the left-hand side, and the analytical K-theory of the reduced group $C^{*}$-algebra on the right-hand side. The computation of the LHS is made possible by the existence of a 3-dimensional model for $\underline{E}G$, which allows to replace equivariant K-homology by Bredon homology. We pay due attention to the presence of torsion in $G$, leading to an extensive study of the wallpaper groups associated with finite subgroups. For the first and third groups, the computations in $K_{0}$ provide explicit generators that are matched by the Baum–Connes assembly map.

show abstract

Exponential growth rates of free and amalgamated products

Cited by 7 publications

References 19 publications

Minimal exponential growth rates of metabelian Baumslag–Solitar groups and lamplighter groups

Minimal exponential growth rates of metabelian Baumslag–Solitar groups and lamplighter groups

Cubic graphs and the golden mean

Equivariant K-Homology and K-Theory for Some Discrete Planar Affine Groups

Contact Info

Product

Resources

About