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The University of Southampton
Mathematical Sciences

Research project: Boundary representations of groups

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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.

The starting point for these boundary/representation constructions draws on the spirit of geometric group theory: having a discrete group acting on a "nice" space (e.g. hyperbolic or CAT(0)). This yields an action of the group on the boundary of that nice space. En route to analytic methods, the boundary is endowed with a structure of a metric measure space.

The current research concerns generalisations of Figa-Talamanca and Picardello's constructions:

  • Down the "metric" rabbit-hole towards representations and proper actions of hypebolic, relatively hyperbolic and acylindrically hyperbolic groups. The overall goal is to construct a family of uniformly bounded representations which approximates the trivial representation (a.k.a. Shalom's conjecture). This is an ambitious goal, and we focus on computing with concrete examples, e.g. Sp(n,ℤ).
  • Keeping the "combinatorics" in the picture: The focus is on the boundaries and uniformly bounded representations in situations where one can use combinatorics in the "nice" space: for example CAT(0) cube complexes, or Cayley graphs of small cancellation groups.

Related research groups

Pure Mathematics
Geometric Group Theory
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