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University of Oklahoma

1. Soroko, Ignat. Dehn functions of subgroups of right-angled Artin groups.

Degree: PhD, 2018, University of Oklahoma

URL: http://hdl.handle.net/11244/299687

The question of what is a possible range for the Dehn functions (a.k.a. isoperimetric spectrum) for certain classes of groups is a natural and interesting one. Due to works of many authors starting with Gromov, we know a lot about the isoperimetric spectrum for the class of all finitely presented groups. Much less is known for other natural classes of groups, such as subgroups of CAT(0) groups or of right-angled Artin groups. The isoperimetric spectrum for the subgroups of right-angled Artin groups, known so far, consists of polynomials up to degree 4 and exponential functions. We extend the knowledge of this spectrum to contain the set of all positive integers. We start by constructing a series of free-by-cyclic groups whose monodromy automorphisms grow as n^k, which admit a virtual embedding into suitable right-angled Artin groups. As a consequence we produce examples of right-angled Artin groups containing finitely presented subgroups whose Dehn functions grow as n^(k+2).
*Advisors/Committee Members: Brady, Noel (advisor), Heyck, Hunter (committee member), Forester, Max (committee member), Schmidt, Ralf (committee member), Tao, Jing (committee member).*

Subjects/Keywords: Dehn functions; right-angled Artin groups; special cube complexes; free-by-cyclic groups

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Soroko, I. (2018). Dehn functions of subgroups of right-angled Artin groups. (Doctoral Dissertation). University of Oklahoma. Retrieved from http://hdl.handle.net/11244/299687

Chicago Manual of Style (16^{th} Edition):

Soroko, Ignat. “Dehn functions of subgroups of right-angled Artin groups.” 2018. Doctoral Dissertation, University of Oklahoma. Accessed January 28, 2021. http://hdl.handle.net/11244/299687.

MLA Handbook (7^{th} Edition):

Soroko, Ignat. “Dehn functions of subgroups of right-angled Artin groups.” 2018. Web. 28 Jan 2021.

Vancouver:

Soroko I. Dehn functions of subgroups of right-angled Artin groups. [Internet] [Doctoral dissertation]. University of Oklahoma; 2018. [cited 2021 Jan 28]. Available from: http://hdl.handle.net/11244/299687.

Council of Science Editors:

Soroko I. Dehn functions of subgroups of right-angled Artin groups. [Doctoral Dissertation]. University of Oklahoma; 2018. Available from: http://hdl.handle.net/11244/299687

University of Oklahoma

2. Stucky, Ben. Cubulating one-relator products with torsion.

Degree: PhD, 2019, University of Oklahoma

URL: http://hdl.handle.net/11244/319548

Since the resolution of the virtual Haken conjecture in the theory of hyperbolic 3-manifolds, there has been much attention devoted to CAT(0) cube complexes. These non-positively curved metric spaces are powerful tools for understanding infinite, finitely 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 CAT(0) cube complex) is sufficient 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 defining 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 defining relator has exponent at least 4 admits a geometric action on a CAT(0) cube complex if the factors do. Our results are framed in the more general context of "staggered" quotients of free products of finitely 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.
*Advisors/Committee Members: Forester, Max (advisor), Jablonski, Michael (committee member), Tao, Jing (committee member), Greene, Scott (committee member), Brady, Noel (committee member).*

Subjects/Keywords: Mathematics; Geometric group theory; Topological methods in group theory; Non-positively curved spaces and groups

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Stucky, B. (2019). Cubulating one-relator products with torsion. (Doctoral Dissertation). University of Oklahoma. Retrieved from http://hdl.handle.net/11244/319548

Chicago Manual of Style (16^{th} Edition):

Stucky, Ben. “Cubulating one-relator products with torsion.” 2019. Doctoral Dissertation, University of Oklahoma. Accessed January 28, 2021. http://hdl.handle.net/11244/319548.

MLA Handbook (7^{th} Edition):

Stucky, Ben. “Cubulating one-relator products with torsion.” 2019. Web. 28 Jan 2021.

Vancouver:

Stucky B. Cubulating one-relator products with torsion. [Internet] [Doctoral dissertation]. University of Oklahoma; 2019. [cited 2021 Jan 28]. Available from: http://hdl.handle.net/11244/319548.

Council of Science Editors:

Stucky B. Cubulating one-relator products with torsion. [Doctoral Dissertation]. University of Oklahoma; 2019. Available from: http://hdl.handle.net/11244/319548