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  • 4 - Discrete subgroups of semisimple Lie groups, beyond lattices

  • Edited by C. M. Campbell, University of St Andrews, Scotland, M. R. Quick, University of St Andrews, Scotland, E. F. Robertson, University of St Andrews, Scotland, C. M. Roney-Dougal, University of St Andrews, Scotland, D. I. Stewart, University of Manchester
  • Book:
    Groups St Andrews 2022 in Newcastle
    Published online:
    21 November 2024
    Print publication:
    12 December 2024, pp 118-190

  • Summary

    Discrete subgroups of $\SL(2,\R)$ are well understood, and classified by the geometry of the corresponding hyperbolic surfaces. Discrete subgroups of higher-rank semisimple Lie groups, such as $\SL(n,\R)$ for $n>2$, remain more mysterious. While lattices in this setting are rigid, there also exist more flexible, ``thinner discrete subgroups, which may have large and interesting deformation spaces, giving rise in particular to so-called higher Teichm\uller theory. We survey recent progress in constructing and understanding such discrete subgroups from a geometric and dynamical viewpoint.