A $2$-complex with contracting non-positive immersions and positive maximal irreducible curvature

TL;DR

This paper constructs a specific 2-complex demonstrating that contracting non-positive immersions and positive maximal irreducible curvature are distinct properties, addressing a question in geometric group theory.

Contribution

It provides a counterexample showing these two properties are not equivalent, clarifying their relationship in the context of 2-complexes.

Findings

The 2-complex has contracting non-positive immersions.

It exhibits positive maximal irreducible curvature.

This distinction answers a question by H. Wilton.

Abstract

We prove that the $2$-complex associated to the presentation $\langle a,b \mid b,bab^{-1}a^{-2}\rangle$ has contracting non-positive immersions and positive maximal irreducible curvature. This example shows that the contracting non-positive immersions property is not equivalent to the notion of non-positive irreducible curvature, answering a question raised by H. Wilton.