Representation Growth of Linear Groups

By Michael Larsen, Alexander Lubotzky. 2006

88 citations recorded by the source corpus.

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 $\calz_\Gamma(s) = \suml^\infty_{n=1} r_n(\Gamma)n^{-s}$. When $\Gamma$ is an arithmetic group satisfying the congruence subgroup property then $\calz_\Gamma(s)$ 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.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7cf4b19a6721fdf41f93eb159b14cbdcce178443

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Citations in the corpus

Representation zeta functions of compact p-adic analytic groups and arithmetic groups (2010, background): “Several of the results of [35] concern the abscissae of convergence of the local representation zeta functions occurring as Euler factors on the right hand side of (1.”
Representation growth and rational singularities of the moduli space of local systems (2013, background): “In [34], the authors prove that α( ) ≥ 1/15.”
Representation zeta functions of nilpotent groups and generating functions for Weyl groups of type B (2011, background): “A variety of tools from complex analysis, algebraic geometry, model theory and combinatorics is available to investigate these zeta functions, and thus to shed light on the arithmetic and asymptotic properties of the sequence (rn(G)); see, for instance, [2, 4, 29].”