Scales for co-compact embeddings of virtually free groups. (English) Zbl 1137.22003
Let \(\Gamma\) be a group which is virtually free of rank at least \(2\) and let \({\mathcal F}_{td}(\Gamma)\) be the family of totally disconnected, locally compact groups containing \(\Gamma\) as a cocompact lattice. A typical representative of \({\mathcal F}_{td}(\Gamma)\) is the group of automorphisms of the Cayley graph of \(\Gamma\) with respect to some finite set of its generators. Given \(G\in {\mathcal F}_{td}(\Gamma)\), the scale function of \(G\) on an automorphism \(g\in \operatorname{Aut}(G)\) is defined by
\[ s_G(g) = \min \{ [g(V) : g(V) \cap V] \mid V \text{ a compact open subgroup of } G \}. \]
It is a measure of the minimal distortion of compact open subgroups of \(G\) under \(g\).
The main result of the paper is that the values of the scale function with respect to the groups in \({\mathcal F}_{td}(\Gamma)\) evaluated on the subset \(\Gamma\) have only finitely many prime divisors. This result can be considered as a part of the program suggested by G. Willis in [Random walks and geometry, 295–316 (2004; Zbl 1057.22005)]. The finiteness of the number of prime factors of \(s_G\) evaluated on \(\Gamma\) for a fixed group \(G \in {\mathcal F}_{td}(\Gamma)\) was known before [G. Willis, Bull. Lond. Math. Soc. 33, 168–174 (2001; Zbl 1020.22002)]; what is new here is the uniform property with respect to the whole family \({\mathcal F}_{td}(\Gamma)\).
The proof uses previous work by L. Mosher, M. Sageev, and K. Whyte and A. Lubotzky.
The last section of the paper contains some explicit bounds on possible values of the scale function on a non-abelian free group.
\[ s_G(g) = \min \{ [g(V) : g(V) \cap V] \mid V \text{ a compact open subgroup of } G \}. \]
It is a measure of the minimal distortion of compact open subgroups of \(G\) under \(g\).
The main result of the paper is that the values of the scale function with respect to the groups in \({\mathcal F}_{td}(\Gamma)\) evaluated on the subset \(\Gamma\) have only finitely many prime divisors. This result can be considered as a part of the program suggested by G. Willis in [Random walks and geometry, 295–316 (2004; Zbl 1057.22005)]. The finiteness of the number of prime factors of \(s_G\) evaluated on \(\Gamma\) for a fixed group \(G \in {\mathcal F}_{td}(\Gamma)\) was known before [G. Willis, Bull. Lond. Math. Soc. 33, 168–174 (2001; Zbl 1020.22002)]; what is new here is the uniform property with respect to the whole family \({\mathcal F}_{td}(\Gamma)\).
The proof uses previous work by L. Mosher, M. Sageev, and K. Whyte and A. Lubotzky.
The last section of the paper contains some explicit bounds on possible values of the scale function on a non-abelian free group.
Reviewer: Mikhail Belolipetsky (Durham)
MSC:
| 22D05 | General properties and structure of locally compact groups |
| 57M07 | Topological methods in group theory |
| 20E08 | Groups acting on trees |
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