Research project: Boundary representations of groups
Free groups are beautiful objects to study from various points of view; this project leverages the ease with which one can do harmonic analysis on them by effectively combinatorial methods. The origins of the story go back to the work of Pytlik, Szwarc; and Figa-Talamanca, Picardello, Mantero, Zappa, in 1980s and 90s: They constructed a holomorphic family of uniformly bounded representations of free groups both on the Cayley graph of the group and on its boundary (the Cantor set); and used the boundary to compute the spectral decomposition of the Laplacian on the Cayley graph.