JournalsjemsVol. 17, No. 4pp. 925–953

Counting arithmetic subgroups and subgroup growth of virtually free groups

  • Amichai Eisenmann

    The Hebrew University of Jerusalem, Israel
Counting arithmetic subgroups and subgroup growth of virtually free groups cover
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Abstract

Let KK be a pp-adic field, and let H=PSL2(K)H=PSL_2(K) endowed with the Haar measure determined by giving a maximal compact subgroup measure 11. Let ALH(x)AL_H(x) denote the number of conjugacy classes of arithmetic lattices in HH with co-volume bounded by xx. We show that under the assumption that KK does not contain the element ζ+ζ1\zeta+\zeta^{-1}, where ζ\zeta denotes the pp-th root of unity over Qp\mathbb{Q}_p, we have

limxlogALH(x)xlogx=q1\lim_{x\rightarrow\infty}\frac{\log AL_H(x)}{x\log x}=q-1

where qq denotes the order of the residue field of KK.

The main innovation of this paper is the proof of a sharp bound on subgroup growth of lattices in HH 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. 925–953

DOI 10.4171/JEMS/522