People
Robert Kropholler Norbert Wiener Assistant Professor
Contact Info:
Tufts University
Department of Mathematics
503 Boston Avenue
BromfieldPearson
Medford, MA 02155
Email @tufts.edu:
robert.kropholler
Personal website
Expertise:
Geometric Group Theory, Topology
Research:
The natural mathematical language in which to describe
symmetries of any system or object is group theory. Symmetry groups
are often finitely generated, i.e. any symmetry can be realised as a
suitable combination of a finite number of basic symmetries
(generators).
Finitely generated groups are a versatile field of study with many
promising avenues. Different aspects of their character are revealed
by using actions on objects endowed with a range of structures. The
action of a group on its Cayley graph with respect to a choice of
finite generating set reveals large scale geometry; actions on
finite dimensional vector spaces carry different information;
actions on finite sets recover information about finite quotients.
There are many other strategies involving spaces with either
topologies or probability measures. In geometric group theory,
actions by isometries on metric spaces are fundamental.
When we have such an action, we wish to use the geometric structure
of the space to illuminate algebraic properties of the group and
vice versa. For instance, if G acts properly and cocompactly on a
metric space X, then G, equipped with the word metric, is
quasiisometric to X. We can use this to discover interesting
properties. For instance, if X is hyperbolic or CAT(0), then G is
finitely presented. Moreover, G has solvable conjugacy problem and
satisfies a linear or quadratic isoperimetric inequality. The fact
that G is finitely presented gives us a finite way to describe an
infinite group. The isoperimetric inequality of a group is a bound
on how quickly trivial elements of the group can be recognised. In
the cases described above, this can be done quickly, but there exist
finitely presented groups whose isoperimetric inequality grows
faster than any computable function.
Since the existence of an action of a group G on a hyperbolic or
CAT(0) space tells us much about the group, there is great incentive
to determine which subgroups of G also admit such an action. All
subgroups will admit a proper action, the trouble comes in proving
that there is a cocompact action.
