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%T Abstract commensurability and quasi-isometry classification of hyperbolic surface group amalgams.
%A Stark, Emily.
%8 2017-04-20
%R http://localhost/files/pg15bs22z
%X Abstract: Let $\mathcal{X}_S$ denote the class of spaces homeomorphic to two
closed orientable surfaces of genus greater than one identified to each other along an
essential simple closed curve in each surface. Let $\mathcal{C}_S$ denote the set of
fundamental groups of spaces in $\mathcal{X}_S$. In this dissertation, we characterize the
abstract commensurability classes in $\mathcal{C}_S$ in terms of the ratio of the Euler
characteristic of the surfaces identified and the topological type of the curves
identified. We characterize which abstract commensurability classes in $\mathcal{C}_S$
contain a maximal element in $\mathcal{C}_S$. We apply our abstract commensurability
classification to prove each group in $\mathcal{C}_S$ is abstractly commensurable to a
right-angled Coxeter group; in particular, we show that two subclasses of groups in
$\mathcal{C}_S$ embed as finite-index subgroups in right-angled Coxeter groups. We
characterize which groups in $\mathcal{C}_S$ are abstractly commensurable to the
right-angled Coxeter groups studied by Crisp--Paoluzzi in \cite{crisppaoluzzi}, and we
exhibit a maximal element within the class of right-angled Coxeter groups for certain
abstract commensurability classes in $\mathcal{C}_S$. We prove that all groups in
$\mathcal{C}_S$ are quasi-isometric by exhibiting a bilipschitz map between the universal
covers of two spaces in $\mathcal{X}_S$. In particular, we prove that the universal covers
of any two such spaces may be realized as isomorphic cell complexes with finitely many
isometry types of hyperbolic polygons as cells.; Thesis (Ph.D.)--Tufts University, 2015.; Submitted to the Dept. of Mathematics.; Advisor: Genevieve Walsh.; Committee: Ian Biringer, Mauricio Gutierrez, and Kim Ruane.; Keyword: Mathematics.
%[ 2022-10-11
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%~ Tufts Digital Library
%W Institution