Boundaries of CAT(0) groups with isolated flats
Ben-Zvi, Michael.
2019
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In \cite(Gromov87), Gromov introduced hyperbolic groups as a way of
capturing negative curvature. This proved to be a wildly interesting class of
groups with many desirable properties. Since then, work has been done to
generalize results in the hyperbolic setting classes of groups with capture
non-positive curvature. For us the groups which capture non-positive curvature are
$\cat(0)$ groups. ... read moreIn the 90s, works of Bestvina-Mess \cite(BestvinaMess), Bowditch
\cite(BowCutPoints), and Swarup \cite(Swarup) show that all 1-ended hyperbolic
groups have locally connected visual boundaries and therefore path connected ones
as well. Because local connectivity in the boundary fails in even very elementary
examples of $\cat(0)$ group, it is natural to ask when the weaker condition of
path connectivity holds. We address this question in two different ways. The first
has to do with $\cat(0)$ spaces which decompose in a special way. We show that if
$X$ has a \textit(block decomposition) with blocks which have path connected
boundaries, along with a technical condition on certain rays in the space, then we
can conclude that $\bnd X$ is path connected. This can be viewed as a combination
theorem, especially when $X$ admits a geometric group action by $G$ and $G$ splits
as an amalgamated product. The second part of the thesis addresses a special class
of groups: $\cat(0)$ groups with isolated flats. We prove that all such groups
have locally connected visual boundaries. Work of Hruska-Ruane \cite(HR17)
provides necessary and sufficient conditions as to when such a group has a locally
connected boundary. We show that even without local connectivity, these groups
have path connected boundaries.
Thesis (Ph.D.)--Tufts University, 2019.
Submitted to the Dept. of Mathematics.
Advisor: Kim Ruane.
Committee: Mike Mihalik, Genevieve Walsh, and Rob Kropholler.
Keywords: Theoretical mathematics, and Mathematics.read less - ID:
- xp68kv578
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