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Quasi-isometric rigidity of a product of lattices, and coarse geometry of non-transitive graphs

Oh, Josiah
http://rave.ohiolink.edu/etdc/view?acc_num=osu1650590308337838

Abstract Details

2022, Doctor of Philosophy, Ohio State University, Mathematics.
In this dissertation we accomplish two things. 1. We demonstrate quasi-isometric rigidity for the product $L \times \Lambda$, where $L$ is a lattice in a simply connected nilpotent Lie group and $\Lambda$ is a non-uniform lattice in the isometry group of a negatively-curved symmetric space. We show that if $\Gamma$ is a finitely generated group quasi-isometric to $L \times \Lambda$, then up to finite noise, $\Gamma$ is an extension of a non-uniform rank one lattice by a lattice in a simply connected nilpotent Lie group. Under additional hypotheses we further show that this extension is nilcentral, a notion which generalizes central extensions to extensions by a nilpotent group. 2. We study the large-scale geometry of non-transitive graphs. In particular we introduce a geometric generalization of vertex-transitivity for graphs, called coarse transitivity, and we prove that the classical Freudenthal-Hopf theorems on the ends of finitely generated groups hold more generally for coarsely transitive graphs. We also prove that for each finitely generated group $G$, there exist uncountably many regular graphs, no two of which are quasi-isometric, that share several geometric properties with $G$.
Jean-Francois Lafont (Advisor)
Michael Davis (Committee Member)
Jingyin Huang (Committee Member)
Michael Durand (Committee Member)
69 p.
Mathematics
geometric group theory, quasi-isometry, quasi-isometric rigidity, lattice, rank one Lie group, simply connected nilpotent Lie group, transitive graph, growth rate, number of ends, asymptotic dimension

Recommended Citations

Citations

  • Oh, J. (2022). Quasi-isometric rigidity of a product of lattices, and coarse geometry of non-transitive graphs [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1650590308337838

    APA Style (7th edition)

  • Oh, Josiah. Quasi-isometric rigidity of a product of lattices, and coarse geometry of non-transitive graphs. 2022. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1650590308337838.

    MLA Style (8th edition)

  • Oh, Josiah. "Quasi-isometric rigidity of a product of lattices, and coarse geometry of non-transitive graphs." Doctoral dissertation, Ohio State University, 2022. http://rave.ohiolink.edu/etdc/view?acc_num=osu1650590308337838

    Chicago Manual of Style (17th edition)

osu1650590308337838
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This open access ETD is published by The Ohio State University and OhioLINK.