Boundary of CAT(0) Groups With Right Angles.
Abstract: Croke and Kleiner present a torus complex whose universal cover has
a non-locally connected visual boundary. They show that changing the intersection angle of
the gluing loops in the middle torus changes the topological type of the visual boundary.
In this thesis we study the effect on the topology of the boundary if the angle is fixed at
π/2 but the lengths of the π1-generating loop... read mores are
changed. In particular, we investigate the topology of the set of geodesic rays with
infinite itineraries. We identify specific infinite-itinerary geodesics whose corresponding
subsets of the visual boundary change their topological type under the length change in the
space. This is a constructive and explicit proof of a result contained in Croke and
Kleiner's more general theorem in a later paper. The construction and view point of this
proof is crucial to proving the next result about Tits boundary: we show that the Tits
boundaries of Croke Kleiner spaces are homeomorphic under lengths variation. Whether this
is true for the general set of CAT(0) 2-complexes is still open. We also study the
invariant subsets of the set of geodesic rays with infinite itineraries. In the next
chapter, we consider the geometry of actions of right-angled Coxeter groups on the
Croke-Kleiner space. We require the group act cocompactly, properly discontinuously and by
isometries and determine that the resulting gluing angles of the loops must be $pi;/2.
Together with the first result we show that right-angled Coxeter groups does not have
unique G-equivariant visual boundaries. This study aims to contribute to the investigation
of whether right-angled Coxeter groups have unique boundaries. Lastly, we begin the study
of uniqueness of right-angled Coxeter groups that acts geometrically on a given CAT(0)
space. We start the project by letting right-angled Coxeter groups at on a regular,
infinite, 4-valence tree and provide a geometrical proof that if a right-angled Coxeter
group acts geometrically on Tr4, then it is an amalgamated product
of finite copies of groups of two types. The last the result is the beginning of the
project that aims to determine all the right-angled Coxeter groups that can act
geometrically on any given space, for instance the Croke-Kleiner space.
Thesis (Ph.D.)--Tufts University, 2013.
Submitted to the Dept. of Mathematics.
Advisor: Kim Ruane.
Committee: Kim Ruane, Mauricio Gutierrez, Genevieve Walsh, and Ruth Charney.
Keyword: Mathematics.read less