Graph products as hierarchically hyperbolic groups

Abstract: Given a finite simplicial graph with a finitely generated group associated to each vertex, the graph product is defined by taking the free product of the vertex groups and adding commutation relations between elements belonging to vertex groups that are connected by a edge in the graph. Common examples of graph products include right-angled Artin groups (where all vertex groups are Z) and right-angled Coxeter groups (where all vertex groups are Z/2Z). Behrstock, Hagen, and Sisto showed that right-angled Artin groups exhibit a notion of non-positive curvature called hierarchical hyperbolicity, with deep geometric consequences such as a Masur-Minsky style distance formula, finite asymptotic dimension, and acylindrical hyperbolicity. By developing analogues of the cubical techniques employed by Behrstock-Hagen-Sisto, we are able to generalise their result, showing that any graph product with hierarchically hyperbolic vertex groups is itself a hierarchically hyperbolic group. In doing so, we answer two questions of Behrstock-Hagen-Sisto and two questions of Genevois. This is joint work with Jacob Russell.

algebraic topologydifferential geometrygeneral topologygroup theorygeometric topology

Audience: researchers in the topic

Ohio State Topology and Geometric Group Theory Seminar

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