Groups Quasi-isometric to H x R^n.
Eisenberg, Andy.
2015
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Abstract: We describe a conjectural characterization of all groups
quasi-isometric to $H \times \R^n$, where $H$ is any non-elementary hyperbolic group, and
we provide an outline of the steps required to establish such a characterization. We carry
out several steps of this plan. We consider those lines $\widehat{L}$ in the asymptotic
cone $\Cone_{\omega}(H)$ which, in a precise sense, ``arise ... read morefrom lines $L$ in $H$''. We
give a complete description of such lines, showing (in particular) that they are extremely
rare in $\Cone_{\omega}(H)$. Given a top-dimensional quasi-flat in $H \times \R^n$, we show
the induced bi-Lipschitz embedded flat in $\Cone_{\omega}(H \times \R^n)$ must lie
uniformly close to some $\widehat{L} \times \R^n$, where $\widehat{L}$ is one of these rare
lines. As a result, we conclude that quasi-actions on $H \times \R^n$ must project to
quasi-actions on $H$ and therefore to homeomorphic actions on $\partial H$. Finally, we
show that such an action on $\partial H$ is a convergence action which is uniform if it is
discrete, and we discuss the work that remains to complete the conjectured
characterization.
Thesis (Ph.D.)--Tufts University, 2015.
Submitted to the Dept. of Mathematics.
Advisor: Kim Ruane.
Committee: Genevieve Walsh, Hao Liang, and Chris Hruska.
Keyword: Mathematics.read less - ID:
- b27746712
- Component ID:
- tufts:21406
- To Cite:
- TARC Citation Guide EndNote
