The quasi-isometry invariance of the coset intersection complex

Abbott, Carolyn; Martínez-Pedroza, Eduardo

doi:10.2140/agt.2026.26.659

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The quasi-isometry invariance of the coset intersection complex

Carolyn Abbott and Eduardo Martínez-Pedroza

Algebraic & Geometric Topology 26 (2026) 659–698

DOI: 10.2140/agt.2026.26.659

Abstract

For a pair $(G, 𝒫)$ consisting of a group and finite collection of subgroups, we introduce a simplicial $G$ -complex $𝒦 (G, 𝒫)$ called the coset intersection complex. We prove that the quasi-isometry type and the homotopy type of $𝒦 (G, 𝒫)$ are quasi-isometric invariants of the group pair $(G, 𝒫)$ . Classical properties of $𝒫$ in $G$ correspond to topological or geometric properties of $𝒦 (G, 𝒫)$ , such as having finite height, having finite width, being almost malnormal, admitting a malnormal core, or having thickness of order one. As applications, we obtain that a number of algebraic properties of $𝒫$ in $G$ are quasi-isometry invariants of the pair $(G, 𝒫)$ . For a certain class of right-angled Artin groups and their maximal parabolic subgroups, we show that the complex $𝒦 (G, 𝒫)$ is quasi-isometric to the extension graph; in particular, it is quasi-isometric to a tree.

Keywords

quasi-isometries, quasi-isometry invariants, coset intersection complex, subgroup height, subgroup width, finite packing, thickness, RAAGs, right angled Artin groups, extension complex, crossing complex

Mathematical Subject Classification

Primary: 20F65

Secondary: 57M07

References

Publication

Received: 29 August 2024

Revised: 14 January 2025

Accepted: 9 February 2025

Published: 11 February 2026

Authors

Carolyn Abbott
Department of Mathematics Brandeis University Waltham, MA United States

Eduardo Martínez-Pedroza
Department of Mathematics and Statistics Memorial University of Newfoundland St. John’s, NL Canada

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Volume 26, issue 2 (2026)