# Counting arithmetic subgroups and subgroup growth of virtually free groups

Journal of the European Mathematical Society (2015)

- Volume: 017, Issue: 4, page 925-953
- ISSN: 1435-9855

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topEisenmann, 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 -

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