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The 2-torsion in the Farrell--Tate cohomology of PSL(4,Z), and torsion subcomplex reduction via discrete Morse theory

Alexander D. Rahm, Anh Tuan Bui, Matthias Wendt

Abstract

In the present paper, we use discrete Morse theory to provide a new implementation of torsion subcomplex reduction for arithmetic groups. This leads both to a simpler algorithm as well as runtime improvements. To demonstrate the technique, we compute the mod 2 Farrell-Tate cohomology of PSL(4,Z).

The 2-torsion in the Farrell--Tate cohomology of PSL(4,Z), and torsion subcomplex reduction via discrete Morse theory

Abstract

In the present paper, we use discrete Morse theory to provide a new implementation of torsion subcomplex reduction for arithmetic groups. This leads both to a simpler algorithm as well as runtime improvements. To demonstrate the technique, we compute the mod 2 Farrell-Tate cohomology of PSL(4,Z).

Paper Structure

This paper contains 3 sections, 1 theorem, 7 equations, 2 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Torsion subcomplex reduction via discrete Morse theory
  3. Construction of a discrete gradient vector field

Key Result

Theorem 1.1

The mod-2 Farrell--Tate cohomology of $\operatorname{PSL}_4(\mathbb{Z})$ has the following dimensions over ${\mathbb{F}}_2$ in degrees $0 \leq q \leq 6$: In degrees $7 \leq q \leq 42$, the dimension of the Farrell--Tate cohomology $\dim_{{\mathbb{F}}_2}\widehat{\operatorname{H}}^q(\operatorname{PSL}_4(\mathbb{Z}); {\mathbb{F}}_2)$ and hence the dimension of the group cohomology $\dim_{{\mathbb{F}

Figures (2)

  • Figure 1: Example: The 2-torsion subcomplex for SL(3,Z)
  • Figure 2: We can also put an essentially different discrete gradient vector field on the $2$-torsion subcomplex from Figure \ref{['nice-vector-field']}:

Theorems & Definitions (4)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.3
  • Definition 2.4