Euler Characteristics of a Family of Congruence Subgroups of $G{L}_m(\mathbb{Z})$

Abstract

The congruence subgroups ${\Gamma }_1(m,p)$ that we consider here are subgroups of $G{L}_m(\mathbb{Z})$ that fix the vector $(0,\dots ,0,1)\mathrm{mod}p$, where $p\ge 5$ is a prime. We present a method and many computations of homological Euler characteristics of $G{L}_m(\mathbb{Z})$ and ${\Gamma }_1(m,p)$ with coefficients in any highest weight representation $V$. By homological Euler characteristics we mean the alternating dimensions of cohomology of the group with coefficient in $V$. We compute the homological Euler characteristics for ${\Gamma }_1(2,p)$, and ${\Gamma }_1(3,p)$ with coefficients in any finite dimensional highest weight representation. Also we compute the homological Euler characteristics for of ${\Gamma }_1(4,p)$ and ${\Gamma }_1(5,p)$ with coefficients in the trivial and the determinant representations. We give application to cohomology of ${\Gamma }_1(3,p)$ with trivial and with determinant representation. We also give an alternative method for computing the cohomology of $G{L}_4(\mathbb{Z})$ compared to \cite{GL4}. The methods in this paper are a continuation of result from \cite{Thesis, EulerChar}.

31 pages, accepted