On the integral homology of PSL4(Z) and other arithmetic groups

TL;DR

This paper computes the integral homology of PSL4(Z) up to degree 5 and analyzes its p-part for higher degrees, also applying methods to other arithmetic groups like PGL3(Z[i]) and PGL3(Z[exp(2pi i/3)]).

Contribution

It provides explicit calculations of the integral homology for PSL4(Z) and extends the approach to other arithmetic groups, offering new insights into their homological properties.

Findings

Integral homology of PSL4(Z) in degrees ≤ 5 determined.

p-part of homology for primes p≥5 analyzed in higher degrees.

Homology descriptions for PGL3(Z[i]) and PGL3(Z[exp(2pi i/3)]) in degrees ≤ 5 provided.

Abstract

We determine the integral homology of PSL4(Z) in degrees at most 5 and determine its p-part in higher degrees for the primes p>=5. Our method applies to other arithmetic groups; as illustrations we include descriptions of the integral homology of PGL3(Z[i]) and PGL3(Z[\exp(2\pi{i}/3)]) in degrees at most 5.