Subgroups of finite index in Baumslag–Solitar groups

Dudkin, F. A.

doi:10.1007/s10469-010-9091-8

Cited by 9 publications

(8 citation statements)

References 5 publications

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“…Then C m m,∞ is the Ellis group of the action (X m , G m , Φ m ), with an associated group chain {G n } n m . In particular, C 0 0,∞ = C ∞ , defined by (15). Since G k ⊂ G m , by definition we have that C m k,∞ ⊂ C m m,∞ , and so C m k,∞ is a clopen neighborhood of the identity in C m m,∞ .…”

Section: The Asymptotic Discriminantmentioning

confidence: 99%

“…Proof Consider the subgroups

C_{n}^{m}

G_{n}

. By , a finite index subgroup in

B S (1, q)

can be written down as

C_{n}^{m} = false⟨ τ^{ℓ_{m, n}}, σ^{k_{m, n}} τ^{r_{m, n}} false⟩,

for some values

r_{m, n}, k_{m, n}

and

ℓ_{m, n}

, where

0 ⩽ r_{m, n} ⩽ ℓ_{m, n} - 1

, and

ℓ_{m, n}

is coprime to

q

. That is,

C_{n}^{m}

is generated by two elements, which we now compute.…”

Section: Stable Arboreal Representationsmentioning

confidence: 99%

“…In this section we make a computation similar to the ones in [15,Section 1] in order to determine when a finite index subgroup S of BS (1, q) is normal in a proper subgroup H of BS (1, q). The divisibility conditions of [15] can be obtained from ours by setting H = BS(1, q), up to a sign.…”

Section: Appendix Subgroups Of Finite Index In the Baumslag-solitar mentioning

confidence: 99%

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Arboreal Cantor actions

Lukina

2018

J. London Math. Soc.

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In this paper, we consider minimal equicontinuous actions of discrete countably generated groups on Cantor sets, obtained from the arboreal representations of absolute Galois groups of fields. In particular, we study the asymptotic discriminant of these actions. The asymptotic discriminant is an invariant obtained by restricting the action to a sequence of nested clopen sets, and studying the isotropies of the enveloping group actions in such restricted systems. An enveloping (Ellis) group of such an action is a profinite group. A large class of actions of profinite groups on Cantor sets is given by arboreal representations of absolute Galois groups of fields. We show how to associate to an arboreal representation an action of a discrete group, and give examples of arboreal representations with stable and wild asymptotic discriminant.

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Section: The Asymptotic Discriminantmentioning

confidence: 99%

“…Proof Consider the subgroups

C_{n}^{m}

G_{n}

. By , a finite index subgroup in

B S (1, q)

can be written down as

C_{n}^{m} = false⟨ τ^{ℓ_{m, n}}, σ^{k_{m, n}} τ^{r_{m, n}} false⟩,

for some values

r_{m, n}, k_{m, n}

and

ℓ_{m, n}

, where

0 ⩽ r_{m, n} ⩽ ℓ_{m, n} - 1

, and

ℓ_{m, n}

is coprime to

q

. That is,

C_{n}^{m}

is generated by two elements, which we now compute.…”

Section: Stable Arboreal Representationsmentioning

confidence: 99%

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Arboreal Cantor actions

Lukina

2018

J. London Math. Soc.

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show abstract

“…In particular, in a non-ascending deformation space any two reduced graphs have the same number of vertices and edges, which might be no longer true in the ascending case. We refer the reader to [22] for a simple example (in the case of BS (2,6)). Note that if G is a GBS group with no non-trivial integral moduli, and H is a finite index subgroup of G, then H also has no non-trivial integral moduli, by Remark 2.2.…”

Section: Preliminariesmentioning

confidence: 99%

“…First results on commensurability of (generalised) Baumslag-Solitar groups are closely linked to the study of their quasi-isometric classification, see [9,28]. In [28] (see also [6]), Whyte described finite index subgroups of BS(m, n), where gcd(m, n) = 1 and showed that no two groups in this class are commensurable. In his work, see [20], Levitt studied the class of GBS groups that generalises the condition gcd(m, n) = 1 for BS(m, n), that is the class of GBS groups without proper plateaus.…”

Section: Introductionmentioning

confidence: 99%