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Robert Kropholler
Norbert Wiener Assistant Professor
Robert Kropholler
Contact Info:
Tufts University
Department of Mathematics
503 Boston Avenue
Bromfield-Pearson
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 quasi-isometric 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.