# Representation growth of linear groups

Michael Larsen; Alexander Lubotzky

Journal of the European Mathematical Society (2008)

- Volume: 010, Issue: 2, page 351-390
- ISSN: 1435-9855

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topLarsen, Michael, and Lubotzky, Alexander. "Representation growth of linear groups." Journal of the European Mathematical Society 010.2 (2008): 351-390. <http://eudml.org/doc/277646>.

@article{Larsen2008,

abstract = {Let $\Gamma $ be a group and $r_n(\Gamma )$ the number of its $n$-dimensional irreducible complex representations. We define and study the associated representation zeta function
$\mathcal \{Z\}_\Gamma (s)=\sum ^\infty _\{n=1\} r_n(\Gamma )n^\{-s\}$. When $\Gamma $ is an arithmetic group satisfying the congruence subgroup property then $\mathcal \{Z\}_\Gamma (s)$ has an “Euler factorization”. The “factor at infinity” is sometimes called the “Witten zeta function” counting the rational representations of an algebraic group. For these we determine precisely the abscissa of convergence. The local factor at a finite place counts the finite representations of suitable open subgroups $U$ of the associated simple group $G$ over the associated local field $K$. Here we show a surprising dichotomy: if $G(K)$ is compact (i.e. $G$ anisotropic over $K$) the abscissa of convergence goes to 0 when $\dim G$ goes to infinity, but for isotropic groups it
is bounded away from $0$. As a consequence, there is an unconditional positive lower bound for the abscissa for arbitrary finitely generated linear groups. We end with some observations
and conjectures regarding the global abscissa.},

author = {Larsen, Michael, Lubotzky, Alexander},

journal = {Journal of the European Mathematical Society},

keywords = {zeta functions; group representations; arithmetic groups; Zeta functions; group representations; arithmetic groups},

language = {eng},

number = {2},

pages = {351-390},

publisher = {European Mathematical Society Publishing House},

title = {Representation growth of linear groups},

url = {http://eudml.org/doc/277646},

volume = {010},

year = {2008},

}

TY - JOUR

AU - Larsen, Michael

AU - Lubotzky, Alexander

TI - Representation growth of linear groups

JO - Journal of the European Mathematical Society

PY - 2008

PB - European Mathematical Society Publishing House

VL - 010

IS - 2

SP - 351

EP - 390

AB - Let $\Gamma $ be a group and $r_n(\Gamma )$ the number of its $n$-dimensional irreducible complex representations. We define and study the associated representation zeta function
$\mathcal {Z}_\Gamma (s)=\sum ^\infty _{n=1} r_n(\Gamma )n^{-s}$. When $\Gamma $ is an arithmetic group satisfying the congruence subgroup property then $\mathcal {Z}_\Gamma (s)$ has an “Euler factorization”. The “factor at infinity” is sometimes called the “Witten zeta function” counting the rational representations of an algebraic group. For these we determine precisely the abscissa of convergence. The local factor at a finite place counts the finite representations of suitable open subgroups $U$ of the associated simple group $G$ over the associated local field $K$. Here we show a surprising dichotomy: if $G(K)$ is compact (i.e. $G$ anisotropic over $K$) the abscissa of convergence goes to 0 when $\dim G$ goes to infinity, but for isotropic groups it
is bounded away from $0$. As a consequence, there is an unconditional positive lower bound for the abscissa for arbitrary finitely generated linear groups. We end with some observations
and conjectures regarding the global abscissa.

LA - eng

KW - zeta functions; group representations; arithmetic groups; Zeta functions; group representations; arithmetic groups

UR - http://eudml.org/doc/277646

ER -

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