Topological generation of simple algebraic groups

  • Timothy C. Burness

    University of Bristol, Bristol, UK
  • Spencer Gerhardt

    University of Southern California, Los Angeles, USA
  • Robert M. Guralnick

    University of Southern California, Los Angeles, USA
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Abstract

Let be a simple algebraic group over an algebraically closed field and let be an irreducible subvariety of  with . In this paper, we consider the general problem of determining if there exists a tuple such that is Zariski dense in . We are primarily interested in the case where and each is a conjugacy class of  comprising elements of prime order modulo the center of . In this setting, our main theorem gives a complete solution to the problem when is a symplectic or orthogonal group. By combining our results with earlier work on linear and exceptional groups, this gives an almost complete solution for all simple algebraic groups. We also present several applications. For example, we use our main theorem to show that many faithful representations of symplectic and orthogonal groups are generically free. We also establish new asymptotic results on the probabilistic generation of finite simple groups by pairs of prime order elements, completing a line of research initiated by Liebeck and Shalev over 25 years ago.

Cite this article

Timothy C. Burness, Spencer Gerhardt, Robert M. Guralnick, Topological generation of simple algebraic groups. J. Eur. Math. Soc. (2024), published online first

DOI 10.4171/JEMS/1425