Counting arithmetic subgroups and subgroup growth of virtually free groups

Abstract

Let $K$ be a $p$-adic field, and let $H=PS{L}_2(K)$ endowed with the Haar measure determined by giving a maximal compact subgroup measure $1$. Let $A{L}_H(x)$ denote the number of conjugacy classes of arithmetic lattices in $H$ with co-volume bounded by $x$. We show that under the assumption that $K$ does not contain the element $\zeta +{\zeta }^{-1}$, where $\zeta$ denotes the $p$-th root of unity over ${\mathbb{Q}}_p$, we have

where $q$ denotes the order of the residue field of $K$.

The main innovation of this paper is the proof of a sharp bound on subgroup growth of lattices in $H$ as above.