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

Abstract

We prove that the $2$-complex associated to the presentation $⟨a,b\mid b,ba{b}^{-1}{a}^{-2}⟩$ 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.