# Higher generating subgroups and Cohen-Macaulay complexes

## TL;DR

This paper develops methods to identify higher generating families of subgroups for groups acting on Cohen-Macaulay complexes, with applications to groups with BN-pairs and automorphism groups of free groups.

## Contribution

It introduces a new approach to find higher generating subgroups in Cohen-Macaulay complexes and applies it to groups with BN-pairs and automorphism groups of free groups.

## Key findings

- Higher generating families of subgroups are constructed for groups acting on Cohen-Macaulay complexes.
- Application to groups with BN-pairs shows higher generation by parabolic and Levi subgroups.
- Describes higher generating families of parabolic subgroups in Aut(F_n).

## Abstract

We show how to find higher generating families of subgroups, in the sense of Abels and Holz, for groups acting on Cohen-Macaulay complexes. We apply this to groups with a BN-pair to prove higher generation by parabolic and Levi-subgroups and describe higher generating families of parabolic subgroups in Aut(F_n).