Since the resolution of the virtual Haken conjecture in the theory of hyperbolic 3-manifolds, there has been much attention devoted to CATp0q cube complexes. These non-positively curved metric spaces are powerful tools for understanding innite, nitely generated groups in part because of their cubical combinatorics. Simply knowing that a group is cubulable (acts geometrically properly and cocompactly by isometries on a CATp0q cube complex) is sucient to unlock a good deal of structural information about it, and cubulating groups has become an important goal of modern geometric group theory.In 2013, Lauer and Wise showed that a one-relator group with torsion whose dening relator has exponent at least 4 is cubulable. To achieve this, they build a system of nicely-behaved codimension-1 subspaces (walls) in the universal cover and invoke a construction due to Sageev.In this thesis, we achieve a generalization of this result to one-relator products with torsion, namely, that a one-relator product of locally indicable groups whose dening relator has exponent at least 4 admits a geometric action on a CATp0q cube complex if the factors do. Our results are framed in the more general context of staggered quotients of free products of nitely many locally indicable and cubulable groups. The main tools are geometric small-cancellation results for van Kampen diagrams over these groups, which allow us to argue that walls are plentiful and geometrically well-behaved in the universal cover. Relative hyperbolicity of these one-relator products and relative quasiconvexity of wall stabilizers both play a central role. Using Agol's theorem that a hyperbolic, cubulable group is virtually special, we obtain as a corollary that the one-relator products we consider are virtually special provided that the factors are hyperbolic in addition to the other assumptions.xi
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.