Relative sectional curvature in compact angled 2-complexes.

摘要

We define the notion of relative sectional curvature for 2-complexes, and prove that a compact angled 2-complex that has negative sectional curvature relative to planar sections has coherent fundamental group. We analyze a certain type of 1-complex that we call flattenable graphs Gamma→ X for an compact angled 2-complex X, and show that if X has nonpositive sectional curvature, and if for every flattenable graph pi1 (Gamma) → pi1( X) is finitely presented, then X has coherent fundamental group. Finally we show that if X is a compact angled 2-complex with negative sectional curvature relative to pi-gons and planar sections then pi1(X) is coherent. Some results are provided which are useful for creating examples of 2-complexes with these properties, or to test a 2-complex for these properties.