Conformal dimension bounds for certain Coxeter group Bowditch boundaries

TL;DR

The paper bounds the conformal dimension of Bowditch boundaries for large-type Coxeter groups with complete defining graphs by embedding Gromov round trees into the Davis–Moussong complex to obtain lower bounds, and by constructing a geometrically finite CAT($-1$) model space to bound the boundary’s Hausdorff dimension from above. Consequently, it proves there are infinitely many quasi-isometry classes within each family with uniformly bounded edge labels, and it derives corollaries for hyperbolic Coxeter groups with Pontryagin-sphere boundaries, including a density result for conformal dimensions in $(1,\infty)$. The work combines combinatorial round-tree techniques with explicit CAT($-1$) geometry and careful itinerary-based orbit counting to relate geometric growth to analytic invariants of the boundary. These results illuminate the quasi-isometry landscape of non-hyperbolic Coxeter groups and demonstrate how boundary geometry governs large-scale group structure, while also posing open questions about finer classifications and potential alternative model geometries.

Abstract

We give upper and lower bounds on the conformal dimension of the Bowditch boundary of a Coxeter group with defining graph a complete graph and edge labels at least three. The lower bounds are obtained by quasi-isometrically embedding Gromov's round trees in the Davis complex. The upper bounds are given by exhibiting a geometrically finite action on a CAT(-1) space and bounding the Hausdorff dimension of the visual boundary of this space. Our results imply that there are infinitely many quasi-isometry classes within each infinite family of such Coxeter groups with edge labels bounded from above. As an application, we prove there are infinitely many quasi-isometry classes among the family of hyperbolic groups with Pontryagin sphere boundary. Combining our results with work of Bourdon--Kleiner proves the conformal dimension of the boundaries of hyperbolic groups in this family achieves a dense set in $(1,\infty)$.

Figures (12)

• Figure 3.1: The $2$-complex $A_{\boldsymbol{a}_n}$ in the case that $m_{ij}=3$ and the $2$-cells in the Davis--Moussong complex are hexagons. The initial complex $A_0$ is the union of the three hexagons at the top of the figure, and its outer edge path $E_0$ is drawn in red. The outer edge path of $A_{\boldsymbol{a}_n}$ is $E_{\boldsymbol{a}_n}$, which is drawn in red, and its adjacent path $E_{\boldsymbol{a}_{n-1}}$ is drawn in orange. The internal vertices $v_1, \ldots, v_k$ along $E_{\boldsymbol{a}_n}$ are indicated with black circles. The complex is extended by adding $V$ strips of polygons glued to $E_{\boldsymbol{a}_n}$ between the new edges emanating from the internal vertices. (As drawn, $V=3$.)
• Figure 3.2: Gluing in polygons during the inductive step. The red line depicts a segment of the outer edge path $E_{a_n}$.
• Figure 3.3: Configurations of walls locally.
• Figure 4.1: On the left a truncated block is the indicated subspace of the ideal tetrahedron $\mathcal{T}$ with boundary a triangle, three pentagonal faces, and three shaded kite faces. These kites correspond to the three shaded kites in the portion of the Davis complex illustrated on the right. The union of these kites in the Davis complex form a fundamental domain for the action of the triangle subgroup that stabilizes the plane tiled by hexagons. The length of the geodesic segment connecting $x_0$ to $x_i$ equals $\frac{\log 2}{2}$, as computed in \ref{['lemma:hexside']}.
• Figure 4.2: On the left is a regular tiling of the 2-sphere by triangles with angles $\frac{2\pi}{3}$. On the right is a labeled side of the ideal tetrahedron $\mathcal{T}$ viewed in the upper half space model of the hyperbolic plane.

Theorems & Definitions (111)

• Corollary 1.1
• Theorem 1
• Corollary 1.2
• Theorem 2
• Theorem 3
• Theorem 4
• Corollary 1.3
• Conjecture 1.4
• Definition 2.1
• Definition 2.2