Finiteness properties of cubulated groups

By G. Hruska, Daniel T. Wise. Compositio Mathematica, 2012

82 citations recorded by the source corpus.

Abstract We give a generalized and self-contained account of Haglund–Paulin’s wallspaces and Sageev’s construction of the CAT(0) cube complex dual to a wallspace. We examine criteria on a wallspace leading to finiteness properties of its dual cube complex. Our discussion is aimed at readers wishing to apply these methods to produce actions of groups on cube complexes and understand their nature. We develop the wallspace ideas in a level of generality that facilitates their application. Our main result describes the structure of dual cube complexes arising from relatively hyperbolic groups. Let $H_1,\ldots, H_s$ be relatively quasiconvex codimension-1 subgroups of a group $G$ that is hyperbolic relative to $P_1, \ldots, P_r$. We prove that $G$ acts relatively cocompactly on the associated dual CAT(0) cube complex $C$. This generalizes Sageev’s result that $C$ is cocompact when $G$ is hyperbolic. When $P_1,\ldots, P_r$ are abelian, we show that the dual CAT(0) cube complex $C$ has a $G$-cocompact CAT(0) truncation.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f23a6db3855d2b3fa43aed06241d69f2aa50359e

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Citations in the corpus

Mixed 3-manifolds are virtually special (2012, methodology): “ is a cut-surface for gand has no accidental parabolics. In the case where gis parabolic, it suﬃces to use an axis R for gon a horosphere. 4.1. WallNbd-WallNbd Separation. We now describe a tool from [HW13] for verifying WallNbd-WallNbd Separation in relatively hyperbolic spaces. Deﬁnition 4.2. Let ”
Quasiflats in hierarchically hyperbolic spaces (2017, background): “ow follows from the distance formula, Theorem 1.3. 1.3. Wallspaces. Wallspaces were introduced by Haglund{Paulin [HP98] and there are now numerous variants of the notion, elaborated by Hruska{Wise in [HW14]. Here, we recall the relevant denitions for Section 2. See, e.g., [HW14] for more background ”
Weak hyperbolicity of cube complexes and quasi‐arboreal groups (2011, background): “Discussions of Sageev’s construction in a general wallspace setting appear in [CN05], [Nic04] and [HW10].”