Small cancellation groups are bi-exact

TL;DR

The paper proves that finitely generated $C'(rac{1}{33})$-groups (not necessarily finitely presented) are bi-exact, extending the catalog of known bi-exact groups. It achieves this by developing two geometric arrays—one governing peripheral contour data and another controlling hyperbolic edge data—and combining them into a global proper array, then transferring to a square-root form to apply bi-exactness criteria. Consequently, every non-virtually cyclic finitely generated $C'(rac{1}{33})$-group is properly proximal and acylindrically hyperbolic, hence non-amenable and bi-exact; this also yields a positive answer to whether acylindrically hyperbolic groups can be properly proximal in this class. Furthermore, the paper extends the results to certain infinitely generated small cancellation groups, either by decomposing into bi-exact pieces or by embedding into a finitely generated bi-exact group, thus broadening the scope of bi-exact groups within geometric group theory.

Abstract

We prove that finitely generated (not necessarily finitely presented) $C'(\frac{1}{33})$-groups are bi-exact. This is a new class of bi-exact groups.

Figures (4)

• Figure 1: Simple geodesic bigon
• Figure 2: Simple geodesic triangles
• Figure 3: Geodesic bigon
• Figure 4: Geodesic triangles with one edge side

Theorems & Definitions (102)

• Theorem 1.1
• Proposition 1.2
• Corollary 1.3
• proof
• Definition 2.1
• Definition 2.2
• Remark 2.3
• Proposition 2.4
• Definition 2.5
• Definition 2.6