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Strict C(6) complexes

Zachary Munro, Daniel T. Wise

TL;DR

This work defines the strict $C(6)$ small-cancellation framework, placing it between nonpositive and negative curvature regimes, and proves that groups acting properly cocompactly on simply-connected strict $C(6)$ complexes are hyperbolic relative to maximal virtually $\ ext{Z}^2$ subgroups of rank $2$. Using geometric walls and a systolization approach, it transfers flat-geometry aspects to a systolic setting and establishes an isolated-flats framework, enabling relative hyperbolicity results. It then develops convex cocompact core theory for (relatively) quasiconvex subgroups, proving the existence of $d_ extbf{f}$-convex cores and cosparse cores in the strict setting, and shows these phenomena can fail without strictness via a careful counterexample. Overall, the paper connects classical small-cancellation theory with systolic geometry to analyze convex cores and relative quasiconvexity in strict $C(6)$-complex actions, providing new structural and geometric tools for relatively hyperbolic groups.

Abstract

We define strict C(n) small-cancellation complexes, intermediate to C(n) and C(n+1), and we prove groups acting properly cocompactly on a simply-connected strict C(6) complex are hyperbolic relative to a collection of maximal virtually free abelian subgroups of rank 2. We study geometric walls in a simply-connected strict C(6) complex, and we use them to prove a convex cocompact (cosparse) core theorem for (relatively) quasiconvex subgroups of strict C(6) groups. We provide an examples showing the convex cocompact core theorem is false without the strict C(6) assumption.

Strict C(6) complexes

TL;DR

This work defines the strict small-cancellation framework, placing it between nonpositive and negative curvature regimes, and proves that groups acting properly cocompactly on simply-connected strict complexes are hyperbolic relative to maximal virtually subgroups of rank . Using geometric walls and a systolization approach, it transfers flat-geometry aspects to a systolic setting and establishes an isolated-flats framework, enabling relative hyperbolicity results. It then develops convex cocompact core theory for (relatively) quasiconvex subgroups, proving the existence of -convex cores and cosparse cores in the strict setting, and shows these phenomena can fail without strictness via a careful counterexample. Overall, the paper connects classical small-cancellation theory with systolic geometry to analyze convex cores and relative quasiconvexity in strict -complex actions, providing new structural and geometric tools for relatively hyperbolic groups.

Abstract

We define strict C(n) small-cancellation complexes, intermediate to C(n) and C(n+1), and we prove groups acting properly cocompactly on a simply-connected strict C(6) complex are hyperbolic relative to a collection of maximal virtually free abelian subgroups of rank 2. We study geometric walls in a simply-connected strict C(6) complex, and we use them to prove a convex cocompact (cosparse) core theorem for (relatively) quasiconvex subgroups of strict C(6) groups. We provide an examples showing the convex cocompact core theorem is false without the strict C(6) assumption.

Paper Structure

This paper contains 5 sections, 41 theorems, 1 equation, 10 figures.

Table of Contents

  1. Introduction
  2. Classical small-cancellation
  3. Strict small-cancellation
  4. Systolization
  5. Convex cocompact cores

Key Result

Theorem A

Let $G$ act properly cocompactly on a simply-connected strict $\Csix$ complex. Then $G$ is hyperbolic relative to a collection of virtually $\mathbf Z^2$ subgroups.

Figures (10)

  • Figure 1: A disc diagram ladder (left) and an annular diagram ladder (right).
  • Figure 2: From the proof of Corollary \ref{['cor:quotientAnnulus']}.
  • Figure 3: Each red component indicates a possible intersection with another honeycomb.
  • Figure 4: The green indicates the cells to be deleted, leaving the components $W(e)$ and $W(e')$.
  • Figure 5: The subgments $A$, $B$, $C$ of $\mathcal{A}$, $\mathcal{B}$, $\mathcal{C}$ contained in $\mathbb{E}^2_\mathbf{hex}$ are highlighted. The orange segments join endpoints of $A$, $B$ and $B$, $C$ and $C$, $A$. Their lengths are uniformly bounded by $K$.
  • ...and 5 more figures

Theorems & Definitions (110)

  • Theorem A: \ref{['thm:relativelyHyperbolic']}
  • Theorem B: \ref{['cor:convexCore']}
  • Theorem C: \ref{['thm:cosparseCore']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5: $\mathrm{C}(n)$
  • Definition 2.6
  • Definition 2.7
  • ...and 100 more