Euler Characteristics of a Family of Congruence Subgroups of $GL_m(\Z)$

TL;DR

This paper develops methods to compute the homological Euler characteristics of certain congruence subgroups of $GL_m( ext{Z})$, providing explicit calculations for various cases and representations, with applications to cohomology.

Contribution

It introduces a new computational approach for homological Euler characteristics of $ ext{GL}_m( ext{Z})$ and its subgroups, extending previous work and covering multiple cases and representations.

Findings

Computed Euler characteristics for $ ext{GL}_2( ext{Z})$, $ ext{GL}_3( ext{Z})$, $ ext{GL}_4( ext{Z})$, and $ ext{GL}_5( ext{Z})$

Provided explicit results for $ ext{Gamma}_1(m,p)$ with various coefficients

Presented an alternative method for cohomology computation of $ ext{GL}_4( ext{Z})$

Abstract

The congruence subgroups $\Gamma_1(m,p)$ that we consider here are subgroups of $GL_m(\Z)$ that fix the vector $(0,\dots,0,1) \mod p$, where $p\geq 5$ is a prime. We present a method and many computations of homological Euler characteristics of $GL_m(\Z)$ 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 $GL_4(\Z)$ compared to \cite{GL4}. The methods in this paper are a continuation of result from \cite{Thesis, EulerChar}.