Combinatorial group theory
My research interests fall into two main categories: (1) representation theory of discrete groups, mainly Kazhdan property and property tau; (2) combinatorial algebra — applications of different combinatorial methods in abstract algebra.
The main part of my research is related to properties T and tau. These properties arise from the representation theory, and they have many applications in combinatorics.
Another part of my research can be broadly described as combinatorial algebra. My research interests are concentrated in the following topics: automorphism groups, Golod-Shafarevich groups, group rings.
- Groups of oscillating intermediate growth (with Pak, I.), Ann. of Math. (2) 177 (2013), no. 3, 1113–1145.
- Hairy graphs and the unstable homology of \(Mod(g,s)\), \(Out(Fn)\) and \(Aut(Fn)\) (with Conant, J. and Vogtmann, K.) . J. Topol. 6 (2013), no. 1, 119–153.
- Subspace arrangements and property,T. Groups Geom. Dyn. 5 (2011), no. 2, 445–477.
- Presentations of finite simple groups: a computational approach (with Guralnick, R. M., Kantor, W. M., and Lubotzky, A.), J. Eur. Math. Soc. (JEMS) 13 (2011), no. 2, 391–458.
- Presentations of finite simple groups: a quantitative approach (with Guralnick, R. M., Kantor, W. M., and Lubotzky A.), J. Amer. Math. Soc. 21 (2008), no. 3, 711–774.
- Symmetric groups and expander graphs, Invent. Math. 170 (2007), no. 2, 327–354.
- Universal lattices and property tau (with Nikolov, N.), Invent. Math. 165 (2006), no. 1, 209–224.