People
Genevieve Walsh Associate Professor
Contact Info:
Tufts University
Department of Mathematics
574 Boston Ave
Collaborative Learning and
Innovation Complex
Room 211G
Medford, MA 02155
Email @tufts.edu:
genevieve.walsh
Phone: 6176274032
Personal site
Expertise:
Hyperbolic manifolds and orbifolds, lowdimensional topology, group actions
Research:
I am a geometric topologist, and I'm interested in problems in both
geometric topology and geometric group theory. I study groups acting
on spaces in a variety of contexts: groups acting on hyperbolic
space with quotient the complement of a knot in \(S^3\), groups acting
on trees, how to make a "good" space for a group to act on, and the
many ways a particular group can act on a particular space. I also
like to understand the geometry of these spaces.
I was trained (if a mathematician can be trained) as a 3manifold
topologist. Work that came out of my thesis showed that hyperbolic
2bridge knot complements are virtually fibered. The relevant point
is that every 2bridge knot complement has a finite cover which is
very nice geometrically: it is the complement of a link of great
circles in \(S^3\). I've studied when a 3manifold has a cover which
contains an embedded incompressible surface, by using eigenspaces of
covering group action. That every closed hyperbolic 3manifold has
such a cover is known as the virtually Haken conjecture. My current
research on knot complements studies the question of
commensurability: When do two manifolds or orbifolds have a common
finitesheeted cover? Commensurability is an equivalence relation on
manifolds and orbifolds which is very rich even when restricted to
knot complements. It tells us a lot about the geometry of the knot
complement. For example, the shape of the cusp of a knot complement
restricts its commensurability class.
Recently, I've been working on some questions about groups generated
by involutions and the type of spaces they can act on. When does a
rightangled Coxeter group act by reflections in hyperbolic space?
When does the automorphism group of a reflection group act on a
CAT(0) space? My approach to these group theoretical questions is
deeply influenced by 3dimensional hyperbolic manifolds and
orbifolds. In turn, geometric group theory informs my research on
manifolds and orbifolds.
