Group Actions on Real Cubings and Limit Groups over Partially Commutative Groups
Montserrat Casals-Ruiz, Ilya Kazachkov

TL;DR
This paper introduces real cubings, generalizes the concept of real trees, and characterizes groups acting on them as subgroups of graph towers, extending classical results on free and surface groups to partially commutative groups.
Contribution
It defines real cubings and characterizes groups acting on them as subgroups of graph towers, extending Rips' theorem and limit group classifications.
Findings
Groups acting on real cubings are subgroups of graph towers.
Characterization of limit groups over partially commutative groups.
Generalization of classical theorems on free and surface groups.
Abstract
We introduce a class of spaces, called real cubings, and study the stucture of groups acting nicely on these spaces. Just as cubings are a natural generalisation of simplicial trees, real cubings can be regarded as a natural generalisation of real trees. Our main result states that a finitely generated group acts nicely (essentially freely and co-specially) on a real cubing if and only if it is a subgroup of a graph tower (a higher dimensional generalisation of -residually free towers and NTQ-groups). It follows that acts freely, essentially freely and co-specially on a real cubing if and only if is a subgroup of the graph product of cyclic and (non-exceptional) surface groups. In the particular case when the real cubing is a tree, it follows that acts freely, essentially freely and co-specially on the real cubing if and only if it is a subgroup of the free…
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Taxonomy
TopicsGeometric and Algebraic Topology · Advanced Operator Algebra Research · Homotopy and Cohomology in Algebraic Topology
