Graduate Student Colloquium
Edgar A. Bering IV
UIC
Meditations on the Torus
Abstract: What are the symmetries of a torus? Taking a topological point of view, we look
for self-homeomorphisms that are homotopically non-trivial, that is, not isotopic to the identity. We arrive at the mapping class group: homeomorphisms modulo isotopy. Puncturing the torus and taking a group theoretic point of view, the appropriate symmetries are automorphisms of the fundamental group, which in this case is the free group on two generators. Once again we quotient out by `obvious' automorphisms, in this case the inner automorphism group, and arrive at the outer automorphism group. Returning to the unpunctured torus, we take an algebraic point of view, and consider the torus as the quotient
of $\mathbb{R}^2$ by an integer lattice $\mathbb{Z}^2$. From this point of view the symmetries of the torus are the linear transformations fixing the integer lattice, $SL(2,\mathbb{Z})$.
These three groups are, for the torus, isomorphic. In general this is not the case, instead the three perspectives outlined are the starting point of three areas of study: mapping class groups, outer automorphism groups, and arithmetic groups, respectively. While distinct there is a close analogy between these three families of groups that drives several areas of current research. Focusing on the motivating example of the torus we will sketch this analogy and point to current frontiers.
Monday October 19, 2015 at 4:00 PM in BH 209