Geometric Group Theory
Professor Leininger’s research is in geometric group theory, a branch of pure mathematics that studies symmetries of spaces, interpreted very broadly, through the use of geometry. In his work, the symmetries involved arise from geometric considerations, especially through surfaces, like the surface of a doughnut, and from graphs. Thus geometry enters as both the source of the symmetries, as well as a tool for studying the group of symmetries. In Fall semester 2016, the Mathematical Sciences Research Institute will host a semester program on geometric group theory, and Professor Leininger will participate in a substantial portion of the activities.
The “mapping class group of a surface” and the “outer automorphism group of a free group” are the symmetry groups where most of Professor Leininger’s research takes place. Specific projects include the following: (1) continuing the study of the notion of “convex cocompactness” in the mapping class group. With Bestvina, Brombert, and Kent, Professor Leininger will look for simpler ways of describing this concept with the hopes that it will shed light on a long standing open problem of Gromov. (2) With Margalit, he will continue to study how long-term behavior of iteration of symmetries of surfaces is reflected in associated 3-dimensional spaces. (3) He will continue his work with Dowdall and Kapovich on “free-by-cyclic groups,” building analogies between the theory developed for surfaces and the more mysterious theory for graphs.