Scales for co-compact embeddings of virtually free groups

Abstract

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 co-compact lattice. We prove that the values of the scale function with respect to groups in ${\mathcal{F}}_{td}(\Gamma )$ evaluated on the subset $\Gamma$ have only finitely many prime divisors. This can be thought of as a uniform property of the family ${\mathcal{F}}_{td}(\Gamma )$.

12 pages; key words: uniform lattice, virtually free group, totally disconnected group, scale function (Error in references corrected in version 2)