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Counting arithmetic subgroups and subgroup growth of virtually free groups

Amichai Eisenmann — 2015

Journal of the European Mathematical Society

Let $K$ be a $p$ -adic field, and let $H = P S L_{2} (K)$ 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 $ζ + ζ^{- 1}$ , where $ζ$ denotes the $p$ -th root of unity over $ℚ_{p}$ , we have $lim_{x \to \infty} \frac{log A L_{H} (x)}{x log x} = q - 1$ where $q$ denotes the order of the residue field of $K$ .

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