Realising all countable groups as quasi-isometry groups

TL;DR

This work addresses the problem of realizing a given countable group $G$ as the quasi-isometry group of a proper geodesic space, showing that for any countable $G$ there exist uncountably many spaces with $\mathrm{QI}(X) \cong G$, and that hyperbolic $G$ admit hyperbolic realizations. The authors introduce and exploit strong QI-rigidity, constructing new strongly QI-rigid spaces as graphs of strongly QI-rigid rank-one symmetric spaces (notably the quaternionic hyperbolic plane $\mathbb{H}\mathbf{H}^2$) and apply a Frucht-style graph-of-spaces framework to encode $G$ into the quasi-isometry group. The key technical contributions include a comprehensive rigidity theory for uniformly hyperbolic graphs of spaces, boundary-stabilizer control via visibility-manifold techniques, and a robust graph-framing (labelings and tag gadgets) that yields uncountably many non-isomorphic examples realizing the same automorphism group. Altogether, the paper provides a broad, geometric method to realize arbitrary countable (and hyperbolic) groups as quasi-isometry groups of explicit geometric objects, advancing understanding of large-scale symmetries in geometric group theory.

Abstract

Given any countable group $G$, we construct uncountably many quasi-isometry classes of proper geodesic metric spaces with quasi-isometry group isomorphic to $G$. Moreover, if the group $G$ is a hyperbolic group, the spaces we construct are hyperbolic metric spaces. We make use of a rigidity phenomenon for quasi-isometries exhibited by many symmetric spaces, called strong quasi-isometric rigidity. Our method involves the construction of new examples of strongly quasi-isometrically rigid spaces, arising as graphs of strongly quasi-isometrically rigid rank-one symmetric spaces.

Figures (5)

• Figure 1: Constructing the tag $T_\sigma$.
• Figure 2:
• Figure 3:
• Figure 4: Blowing up $\Gamma_{G,S}$ to a 3-regular graph $\Gamma_0$, in the case where $S$ is countably infinite.
• Figure 5: Illustration of a section of our graph of spaces $\mathbf{X}_\Gamma$.

Theorems & Definitions (132)

• Theorem 1
• Theorem 2
• Definition 1.1: Quasi-isometry
• Definition 1.2: Quasi-geodesics
• Definition 1.3: Strongly QI-rigid space
• Definition 1.4: Coarse connectedness
• Lemma 1.5
• proof
• Definition 1.6: Gromov product
• Definition 1.7: Thin triangles