Counting arithmetic subgroups and subgroup growth of virtually free groups

Eisenmann, Amichai. "Counting arithmetic subgroups and subgroup growth of virtually free groups." Journal of the European Mathematical Society 017.4 (2015): 925-953. <http://eudml.org/doc/277366>.

@article{Eisenmann2015,
abstract = {Let $K$ be a $p$-adic field, and let $H=PSL_2(K)$ endowed with the Haar measure determined by giving a maximal compact subgroup measure $1$. Let $AL_H(x)$ 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 \[\lim \_\{x\rightarrow \infty \}\frac\{\log AL\_H(x)\}\{x\log x\}=q-1\] where $q$ denotes the order of the residue field of $K$.},
author = {Eisenmann, Amichai},
journal = {Journal of the European Mathematical Society},
keywords = {arithmetic subgroups; counting lattices; subgroup growth; virtually free groups; arithmetic lattices; $p$-adic fields; graph of groups; arithmetic lattices; p-adic fields; virtually free groups; graph of groups},
language = {eng},
number = {4},
pages = {925-953},
publisher = {European Mathematical Society Publishing House},
title = {Counting arithmetic subgroups and subgroup growth of virtually free groups},
url = {http://eudml.org/doc/277366},
volume = {017},
year = {2015},
}

TY - JOUR
AU - Eisenmann, Amichai
TI - Counting arithmetic subgroups and subgroup growth of virtually free groups
JO - Journal of the European Mathematical Society
PY - 2015
PB - European Mathematical Society Publishing House
VL - 017
IS - 4
SP - 925
EP - 953
AB - Let $K$ be a $p$-adic field, and let $H=PSL_2(K)$ endowed with the Haar measure determined by giving a maximal compact subgroup measure $1$. Let $AL_H(x)$ 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 \[\lim _{x\rightarrow \infty }\frac{\log AL_H(x)}{x\log x}=q-1\] where $q$ denotes the order of the residue field of $K$.
LA - eng
KW - arithmetic subgroups; counting lattices; subgroup growth; virtually free groups; arithmetic lattices; $p$-adic fields; graph of groups; arithmetic lattices; p-adic fields; virtually free groups; graph of groups
UR - http://eudml.org/doc/277366
ER -