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Fit systolic groups, exactly

Martín Blufstein, Victor Chepoi, Huaitao Gui, Damian Osajda

TL;DR

The work defines a fitness condition for systolic complexes and proves that uniformly locally finite fit systolic complexes satisfy Yu's Property A, which implies coarse embeddability and exactness for groups acting properly on them. The authors exploit a combinatorial criterion of Špakula–Wright and analyze interval geometry via normal clique paths to control level-set growth. They then establish fitness for several broad group families, including almost-large-type Artin groups, graphical C(3)–T(6) and classical C(6) small cancellation groups, and supply a 3D example to illustrate the reach of the method. Collectively, the results broaden the catalog of exact groups and provide a versatile framework for deducing Property A from interval structure in bridged/systolic settings, with potential scope for further generalizations and applications in coarse geometry and operator algebras.

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 Špakula and Wright.

Fit systolic groups, exactly

TL;DR

The work defines a fitness condition for systolic complexes and proves that uniformly locally finite fit systolic complexes satisfy Yu's Property A, which implies coarse embeddability and exactness for groups acting properly on them. The authors exploit a combinatorial criterion of Špakula–Wright and analyze interval geometry via normal clique paths to control level-set growth. They then establish fitness for several broad group families, including almost-large-type Artin groups, graphical C(3)–T(6) and classical C(6) small cancellation groups, and supply a 3D example to illustrate the reach of the method. Collectively, the results broaden the catalog of exact groups and provide a versatile framework for deducing Property A from interval structure in bridged/systolic settings, with potential scope for further generalizations and applications in coarse geometry and operator algebras.

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 -- small cancellation groups, and finitely presented classical small cancellation groups are exact. We also provide further examples. Our proof relies on a combinatorial criterion for Property~A due to Špakula and Wright.
Paper Structure (19 sections, 24 theorems, 4 equations, 9 figures)

This paper contains 19 sections, 24 theorems, 4 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Related results
  4. Organization
  5. Preliminaries
  6. Graphs
  7. Yu's Property A
  8. Fit systolic complexes
  9. Level sets and fit systolic complexes
  10. Characterizations of fit systolicity
  11. Normal clique paths in bridged graphs
  12. The proof of the Main Theorem
  13. Large-type Artin groups are fit
  14. Graphical $C(3)$--$T(6)$ groups are fit
  15. Graphical small cancellation complexes
  16. ...and 4 more sections

Key Result

Theorem 3.1

FaJaSoCh A graph is bridged if and only if the balls $B_r(A)$ around convex sets $A$ are convex.

Figures (9)

  • Figure 1:
  • Figure 2: A level set (red) in an interval in a systolic complex/bridged graph. Being "fit" means that level sets are "not too large".
  • Figure 3: On the left: a triplane with half-planes glued in a "zipped" way along a sequence of tetrahedra. A $4$-arrowhead is outlined in thick blue. On the right: the link of a vertex $v_i$, a doubled asteroid.
  • Figure 4: A thick hexagon. The edges in the complete subgraphs $U_l$, $U_r$, $D_l$ and $D_r$ are not drawn.
  • Figure 5: Gluing of two thick hexagons in the link of a real vertex.
  • ...and 4 more figures

Theorems & Definitions (55)

  • Definition 2.1: Bridged/systolic
  • Definition 2.2
  • Theorem 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Proposition 3.5
  • proof
  • Claim 3.6
  • proof
  • ...and 45 more