Abstract
Abstract
Quasi-Isometric Properties of Graph Braid Groups
In my thesis I initiate the study of the quasi-isometric properties
of the
2 dimensional graph braid groups. I do this by studying the
behaviour of
flats in the geometric model spaces of the graph braid groups,
which happen
to be CAT(0) cube complexes. I define a quasi-isometric invariant
of these
graph braid groups called the intersection complex. In certain
cases it is
possible to calculate the dimension of this intersection complex
from the
underlying graph of the graph braid group. And I use the dimension
of the
intersection complex to prove that the family of graph braid groups
B_2(K_n)
are quasi-isometrically distinct for all n. I also show that the
dimension
of the intersection complex for a graph braid group takes on every
possible
non-negative integer value.
Table of Contents
Contents
1 Introduction 1
2 Quasi-Isometries 3
3 CAT(0) Cube Complexes 7
3.1 Right Angled Artin Groups . . . . . . . . . . . . . . . . . . .
8
3.2 Graph Braid Groups . . . . . . . . . . . . . . . . . . . . . .
. 9
3.3 Quasi-Flats in CAT(0) Square Complexes . . . . . . . . . . .
10
4 Graph Braid Groups 13
5 Quarter-Plane Complex 28
5.1 Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 34
5.2 Product subcomplexes and posets . . . . . . . . . . . . . . . .
40
5.3 Preservation of Flats . . . . . . . . . . . . . . . . . . . . .
. . 48
6 Maximal Product Sub-Complexes 54
7 The Intersection Complex 62
7.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 67
Bibliography 72
About this Dissertation
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