The proposed research concerns the mapping class group of a surface and related groups. It contains four families of projects. The first concerns the cohomological properties of finite-index subgroups. Specific goals here include proving an analogue of the Borel stability theorem for the mapping class group and proving a sort of "equivariant homological stability" theorem for congruence subgroups of the special linear group. The second family of projects concerns the Picard groups of finite covers of the moduli space of curves. The goal here is to understand the divisibility properties of certain natural line bundles on these finite covers. The third family of projects concerns the Torelli subgroup of the mapping class group, which is the kernel of action of the mapping class group on the first homology group of the surface. The goal here is to clarify the basic cohomological and combinatorial properties of this group and its subgroups. The final family of projects concerns the analogue of the Torelli subgroup in the automorphism group of a free group. The goal here is to adapt tools that have been successful in studying the mapping class group to the setting of the automorphism group of a free group. In particular, analogues of the curve complex will be studied.
The proposed projects concern mapping class groups, which play a key role in many fields of mathematics, ranging from algebraic geometry and low dimensional topology to mathematical physics. The problems which involve the cohomology groups of the mapping class group seek to measure one of the most basic invariants of these groups ? roughly, the k-dimensional cohomology groups count the k-dimensional "holes" in geometric models for the groups. These play an important role in the applications. Another set of problems concern the combinatorics of these groups. This should allow actual concrete calculations within them, facilitating the investigation of the diverse objects with which they interact.
During the three years of the grant, the PI wrote 10 papers on a variety of mathematical topics and delivered 37 invited addresses concerning his research. He also organized 3 international conferences and 1 special session at a meeting of the American Mathematical Society. As far as educational activities go, the PI began a new job at Rice University at the beginning of the grant. During the three year duration of the grant, he began advising three PhD students and one postdoc. The PhD students have been very productive; one of them wrote a paper and gave an invited address at an international conference, and the other two have papers in progress that will be completed within the next year. The oldest of them will graduate in the spring of 2014. Some highlights of the PI's research during the timeframe of the grant are as follows. 1. He proved a 30 year old conjecture of Dennis Johnson that asserted that the Torelli group (a fundamental object in geometry and topology) can be built out of a small number of basic building blocks. 2. He uncovered a new type of pattern in the "cohomology" of certain types of "congruence subgroups". Cohomology is a measurement of higher-dimensional "holes" in a space, and congruence subgroups are ways of encoding symmetries in certain kinds of geometric and number-theoretic objects. 3. Together with Tara Brendle (University of Glasgow) and Dan Margalit (Georgia Tech), he proved a conjecture of Richard Hain which gives generators for the kernel of the hyperelliptic Torelli group. The hyperelliptic Torelli group is a way of encoding certain kinds of symmetries of two-dimensional shapes (like donuts with multiple holes). It plays an important role in algebraic geometry (the study of solutions to polynomial equations). 4. Together with Matthew Day (University of Arkansas), he discovered a new version of the classical Birman exact sequence, which is an important tool for studying two-dimensional shapes.
