Closed subsets of a CAT(0) 2-complex are intrinsically CAT(0)

TL;DR

This paper proves that closed, rectifiably-connected subsets with trivial first homology in a locally-finite CAT(k) 2-complex are themselves complete CAT(k) spaces, extending the understanding of intrinsic geometry in such complexes.

Contribution

It establishes that such subsets inherit the CAT(k) property under the induced path metric, providing new insights into the geometric structure of CAT(0) complexes.

Findings

Closed, rectifiably-connected subsets with trivial first homology are CAT(k) spaces.

These subsets are complete under the induced path metric.

The result applies to locally-finite CAT(k) polyhedral 2-complexes.

Abstract

Let k be at most 0, and let X be a locally-finite CAT(k) polyhedral 2-complex X, each face with constant curvature k. Let E be a closed, rectifiably-connected subset of X with trivial first singular homology. We show that E, under the induced path metric, is a complete CAT(k) space.