Fit systolic groups, exactly

TL;DR

This paper proves that certain systolic complexes with bounded metric intervals have Property A, implying that groups acting on them are exact and boundary amenable, with applications to various classes of finitely presented groups.

Contribution

It establishes that uniformly locally finite fit systolic complexes possess Property A, extending exactness to a broad class of groups including Artin and small cancellation groups.

Findings

Groups acting on fit systolic complexes are exact.

Large-type Artin groups are exact.

Finitely presented small cancellation groups are exact.

Abstract

A systolic complex/bridged graph is fit when its (metric) intervals are "not too large". We prove that uniformly locally finite fit systolic complexes have Yu's Property A. In particular, groups acting properly on such complexes have Property A, (equivalently) they are exact, and (equivalently) they are boundary amenable. As applications we show that groups from a class containing all large-type Artin groups, as well as all finitely presented graphical $C(3)$--$T(6)$ small cancellation groups, and finitely presented classical $C(6)$ small cancellation groups are exact. We also provide further examples. Our proof relies on a combinatorial criterion for Property~A due to \v{S}pakula and Wright.