On universal modular symbols

TL;DR

This paper clarifies the relationship between different approaches to the homology of Steinberg modules and provides a simplified proof of the Solomon-Tits theorem, connecting combinatorics with algebraic K-theory.

Contribution

It establishes a clear relationship between existing works on Steinberg modules and offers a simplified proof of a key theorem in the field.

Findings

Simplified proof of the Solomon-Tits theorem.

Clarified relationship between Lee-Szczarba and Ash-Rudolph approaches.

Connected combinatorics with homology computations of general linear groups.

Abstract

We clarify the relationship between works of Lee-Szczarba and Ash-Rudolph on the homology of the Steinberg module of a linear Tits building. This yields a simple proof of the Solomon-Tits theorem in this special case. We also give a (weak) relationship between this combinatorics and the one studied by van der Kallen, Suslin and Nesterenko to compute the homology of the general linear group with constant coefficients.