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

Ivan Horozov

Abstract

The congruence subgroups $Γ_1(m,p)$ 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 $Γ_1(m,p)$ 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 $Γ_1(2,p)$, and $Γ_1(3,p)$ with coefficients in any finite dimensional highest weight representation. Also we compute the homological Euler characteristics for of $Γ_1(4,p)$ and $Γ_1(5,p)$ with coefficients in the trivial and the determinant representations. We give application to cohomology of $Γ_1(3,p)$ 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}.

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

Abstract

The congruence subgroups that we consider here are subgroups of that fix the vector , where is a prime. We present a method and many computations of homological Euler characteristics of and with coefficients in any highest weight representation . By homological Euler characteristics we mean the alternating dimensions of cohomology of the group with coefficient in . We compute the homological Euler characteristics for , and with coefficients in any finite dimensional highest weight representation. Also we compute the homological Euler characteristics for of and with coefficients in the trivial and the determinant representations. We give application to cohomology of with trivial and with determinant representation. We also give an alternative method for computing the cohomology of compared to \cite{GL4}. The methods in this paper are a continuation of result from \cite{Thesis, EulerChar}.
Paper Structure (14 sections, 48 theorems, 146 equations, 1 table)

This paper contains 14 sections, 48 theorems, 146 equations, 1 table.

Table of Contents

  1. Introduction
  2. Basic properties of Euler characteristics
  3. Key results of the paper
  4. Euler Characteristic and Cohomology of $GL_2(\mathbb{Z})$ and $SL_2(\mathbb{Z})$
  5. Structure theory
  6. Cohomology of $SL_2(\mathbb{Z})$ and $GL_2(\mathbb{Z})$
  7. Boundary cohomology of $SL_2(\mathbb{Z})$ and $GL_2(\mathbb{Z})$
  8. Euler Characteristics and Cohomology of $\Gamma_1(2,p)$
  9. General Setting for Euler Characteristics of $\Gamma_1(m,p)$
  10. Euler characteristics of $\Gamma_1(3,p)$ with coefficients in any highest weight representation
  11. Cohomology of $\Gamma_1(3,p)$ with trivial and with determinant coefficients
  12. Cohomology of $GL_4(\mathbb{Z})$ with trivial and with determinant coefficients
  13. Euler Characteristics of $\Gamma_1(4,p)$ with trivial and with determinant coefficients
  14. Euler Characteristics of $\Gamma_1(5,p)$ with trivial and with determinant coefficients

Key Result

Theorem 1

The summation is over all possible block-diagonal torsion matrices $A \in \Gamma$ satisfying the following conditions:

Theorems & Definitions (52)

  • Theorem 1
  • Lemma 2
  • Lemma 3
  • Corollary 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Theorem 9
  • Theorem 10
  • ...and 42 more