# Linear syzygies, hyperbolic Coxeter groups and regularity

## TL;DR

This paper explores the relationship between the cohomological dimension of Coxeter groups and the regularity of associated Stanley--Reisner rings, providing new bounds and examples for the regularity of certain monomial ideals.

## Contribution

It establishes a connection between geometric group theory and commutative algebra, introduces new bounds on regularity, and constructs examples with arbitrarily high regularity after linear syzygies.

## Key findings

- Virtual cohomological dimension relates to regularity of Stanley--Reisner rings.
- Constructs examples with arbitrarily high regularity after linear syzygies.
- Provides a doubly logarithmic bound on regularity for Cohen--Macaulay ideals.

## Abstract

We show that the virtual cohomological dimension of a Coxeter group is essentially the regularity of the Stanley--Reisner ring of its nerve. Using this connection between geometric group theory and commutative algebra, as well as techniques from the theory of hyperbolic Coxeter groups, we study the behavior of the Castelnuovo--Mumford regularity of square-free quadratic monomial ideals. We construct examples of such ideals which exhibit arbitrarily high regularity after linear syzygies for arbitrarily many steps. We give a doubly logarithmic bound on the regularity as a function of the number of variables if these ideals are Cohen--Macaulay.