Research in Geometry and Topology

The interplay between algebra and geometry is one of the classical themes in mathematics. Traditionally, one studies geometric objects via their symmetries. In geometric group theory the situation is reversed: one starts with a group of symmetries of an algebraic object (for example, another group) and constructs a geometric object with the same symmetries. Here the focus is on the study of symmetries of surfaces. Surprisingly, its geometry is to a large degree governed by hyperbolic geometry that goes back to Gauss, Lobacevski, Poincare and others. The goal is to better understand this phenomenon. The proposer's PhD students and postdoctoral researchers are also involved in parts of this investigation. The most important result about the dynamics of classical mapping classes is the Nielsen-Thurston classification, which explains the behavior of iterates of simple closed curves. The same goal in the case of surfaces of infinite type looks to be much harder, but certainly worthy of a systematic study. The family of groups of proper homotopy equivalences of a locally finite graph is a sister family to the family of big mapping class groups, with many similarities and intriguing differences that the investigator wants to explore. More classically, the investigator wants to build ``towers'' of successive approximations to the classical mapping class group complexes, as well as complexes associated to automorphism groups of free groups, much like the Taylor series. The process of building these towers is called ``disintegration'' and preliminary work shows that it succeeds for the arc and curve complexes and sheds new light on their large-scale geometry. This is yet another attempt to establish that the complexes associated with automorphism groups of free groups have finite asymptotic dimension, perhaps the main outstanding open question about the large-scale geometry of this group. The questions about ergodic measures could have been asked many years ago, but new methods are needed to answer them. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.