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Michael Ben-Zvi
Norbert Wiener Assistant Professor
Michael Ben-Zvi
Contact Info:
Tufts University
Department of Mathematics
503 Boston Avenue
Medford, MA 02155

Email @tufts.edu:
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Geometric group theory

Geometric group theory explores the rich connection between algebraic and geometric properties of a group. This interplay comes in two different ways. The first is viewing a group, G, as a geometric object itself by fixing a generating set. This endows G with the word metric, allowing us to make sense of distance within the group. We can then try to understand algebraic properties of the group through studying the geometry of G with the word metric. Fixing a generating set also lets us draw a picture of G in the form of its Cayley graph and G naturally acts in a nice way on its Cayley graph. This gives rise to the second way of studying the relationship between algebra and geometry: through spaces on which the group acts geometrically. It turns out that when a group acts geometrically on a space, these two objects are coarsely equivalent, letting us make connections between the algebraic properties of the group and the geometric properties of the space.

A major area of study within geometric group theory has been the study of hyperbolic groups. These are groups which act on spaces with negative curvature and they have lots of nice algebraic and geometric properties. A generalization of hyperbolic groups is CAT(0) groups. These are groups which act on spaces with non-positive curvature. My research has been on understanding when certain properties of hyperbolic groups are also properties of CAT(0) groups. In particular, my focus has been on understanding the boundaries, the space at infinity, of such groups.