Let be a -adic field, and let endowed with the Haar measure determined by giving a maximal compact subgroup measure . Let denote the number of conjugacy classes of arithmetic lattices in with co-volume bounded by . We show that under the assumption that does not contain the element , where denotes the -th root of unity over , we have
where denotes the order of the residue field of .
The main innovation of this paper is the proof of a sharp bound on subgroup growth of lattices in as above.
Cite this article
Amichai Eisenmann, Counting arithmetic subgroups and subgroup growth of virtually free groups. J. Eur. Math. Soc. 17 (2015), no. 4, pp. 925–953DOI 10.4171/JEMS/522