TL;DR
This paper demonstrates that some cubulated groups lack a factor system and have non-acylindrical actions on contact graphs, challenging assumptions in hierarchical hyperbolicity theory.
Contribution
It constructs explicit examples of cubulated groups without factor systems and with non-acylindrical contact graph actions, showing limitations of current methods.
Findings
Existence of cubulated groups without factor systems
Examples of cubulated groups with non-acylindrical contact graph actions
Challenges to the universality of hierarchical hyperbolicity methods
Abstract
The primary method for showing that a given cubulated group is hierarchically hyperbolic is by constructing a factor system on the cube complex. In this paper we show that such a construction is not always possible, namely we construct a cubulated group for which the cube complex does not have a factor system. We also construct a cubulated group for which the induced action on the contact graph is not acylindrical.
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Taxonomy
TopicsGeometric and Algebraic Topology · Mathematical Dynamics and Fractals · Homotopy and Cohomology in Algebraic Topology