Conformal dimension bounds, Pontryagin sphere boundaries, and algebraic fibering of right-angled Coxeter groups

TL;DR

The paper develops a framework to generate infinitely many quasi-isometry classes among hyperbolic right-angled Coxeter groups by introducing a graph-theoretic $(n,m)$--branching condition that guarantees the embedding of combinatorial round trees into the Davis complex. This embedding yields explicit lower bounds on the conformal dimension of the boundary and, via Pontryagin-sphere realizations and flag-no-square triangulations, produces groups whose boundaries are Pontryagin spheres with unbounded conformal dimension. Building on LMSSW’s virtual fibering program, the authors extend techniques to show that for every $n \, ext{ge}\, 2$ there exist infinitely many RACGs that virtually algebraically fiber while exhibiting unbounded boundary conformal dimension and fixed virtual cohomological dimension. The work combines round-tree embeddings, L2-cohomology considerations, and detailed constructions from incidence geometries (generalized $m$-gons and Levi graphs of projective/affine/biaffine planes) to produce rich families of RACGs with high conformal-dimension boundaries and multiple quasi-isometry classes, advancing understanding of boundary geometry and fiberings in right-angled Coxeter groups.

Abstract

We introduce a graph-theoretic condition, called $(n,m)$--branching, that ensures a combinatorial round tree with controlled branching parameters can be quasi-isometrically embedded in the Davis complex of the right-angled Coxeter group defined by the graph. This construction yields a lower bound on the conformal dimension of the boundary of such a hyperbolic group. We exhibit numerous families of graphs with this property, including many 1-dimensional spherical buildings. We prove an embedding result, showing that under mild hypotheses a flag-no-square graph embeds as an induced subgraph in a flag-no-square triangulation of a closed surface. We use this to embed our branching graphs into graphs presenting hyperbolic right-angled Coxeter groups with Pontryagin sphere boundary. We conclude there are examples of such groups with conformal dimension tending to infinity, and hence, there are infinitely many quasi-isometry classes within this family. We use conformal dimension to show that recent work of Lafont--Minemyer--Sorcar--Stover--Wells can be upgraded to conclude that for every $n \geq 2$ there exist infinitely many quasi-isometry classes of hyperbolic right-angled Coxeter groups that virtually algebraically fiber and have virtual cohomological dimension $n$.

Figures (8)

• Figure 4.1: An illustration of the $(3,6)$--branching condition. The condition ensures that certain subgraphs, drawn with thickened edges, can be extended to contain induced 6-cycles that intersect only in the original thickened subgraph.
• Figure 4.2: The initial steps of the round tree construction with $(2,6)$--branching. The initial subcomplex $A_0$ consists of two squares incident to the marked vertex. The outer edge path is drawn in blue. At the first blue edge we attach two new squares, colored red and green. At the first interior vertex of the blue path we add four more red squares such that the five red squares plus the square from $A_0$ make a 6--cycle of squares. Similarly add four more green squares to make a second 6--cycle of squares sharing only the $A_0$ square with the red cycle. The $(2,6)$--branching condition makes these choices possible. Continue along the blue path, extending the red and green 'strips' using the branching condition at each successive interior vertex of the blue path.
• Figure 5.1: The replacement simplices for the absolute and relative subdivisions.
• Figure 6.1: Hexagon $H(i,j)$ containing segment $u \mathrel{ } w \mathrel{ } v$.
• Figure 6.2: Hexagon $H(i,j,k)$ containing edge $u \mathrel{ } v$.

Theorems & Definitions (57)

• Theorem : \ref{['thm:Psphere_lower_bound']}
• Theorem : \ref{['thm:confdim_lower_bounds']}
• Theorem : LMSSW-fibering
• Theorem : \ref{['thm:QI_of_fibering_examples']}
• Definition 2.1
• Definition 2.2
• Lemma 2.3
• Definition 2.5
• Definition 2.6
• Theorem 2.7: bourdon-flot, Theorem 1.6.4; buyaloschroeder, Theorem 5.2.17