# The triangle groups (2,4,5) and (2,5,5) are not systolic

## TL;DR

This paper demonstrates that certain hyperbolic triangle groups are not systolic by providing new examples and analyzing their properties, contributing to the understanding of the relationship between hyperbolic and systolic groups.

## Contribution

It introduces new examples of hyperbolic groups that are not systolic and investigates properties of systolic complexes under involutions.

## Key findings

- The triangle groups (2,4,5) and (2,5,5) are not systolic.
- Some subsets of systolic complexes are stable under involutions.
- New insights into the structure of hyperbolic and systolic groups.

## Abstract

In this paper we provide new examples of hyperbolic but nonsystolic groups by showing that the triangle groups $(2,4,5)$ and $(2,5,5)$ are not systolic. Along the way we prove some results about subsets of systolic complexes stable under involutions.