The quasi-isometry invariance of the Coset Intersection Complex

Abstract

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

Version 3. Version accepted by AGT for publication

References in corpus (14)

• Relative hyperbolicity and relative quasiconvexity for countable groups
• Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces
• Embedability between right-angled Artin groups
• Quasiflats in CAT(0) complexes
• Weak hyperbolicity of cube complexes and quasi-arboreal groups
• Packing subgroups in relatively hyperbolic groups
• Residual finiteness, QCERF, and fillings of hyperbolic groups
• Combination of quasiconvex subgroups of relatively hyperbolic groups
• Automorphisms of graph products of groups from a geometric perspective
• Quasi-isometric classification of right-angled Artin groups I: the finite out case
• Quasi-isometric rigidity of subgroups and Filtered ends
• Quasi-isometry invariance of relative filling functions
• Hyperbolically embedded subgroups and quasi-isometries of pairs
• Quasi-isometry invariants of weakly special square complexes