Geometry, Topology and Dynamics Seminar
Subhadip Dey
UC Davis
Hausdorff dimension of the limit sets of Anosov subgroups
Abstract: Patterson-Sullivan measures were introduced by Patterson (1976) and Sullivan (1979)
to study the limit sets of Kleinian groups.
Using these measures, they showed a close relationship between the critical exponent,
$\delta(\Gamma)$, of a Kleinian group $\Gamma < \mathrm{Isom}(\mathbb{H}^n)$ and the
Hausdorff dimension, $\mathrm{Hd}(\Lambda(\Gamma))$, of the limit set
$\Lambda(\Gamma)$ of $\Gamma$. Intuitively, the critical exponent gives a
geometric measurement of the growth rate of a(ny) $\Gamma$-orbit in $\mathbb{H}^n$ and,
on the other hand, the Hausdorff dimension describes the size of the limit set $\Lambda(\Gamma)$.
For instance, for convex-cocompact Kleinian groups $\Gamma$, Sullivan proved that
$\delta(\Gamma) = \mathrm{Hd}(\Lambda(\Gamma))$. Anosov subgroups
(or Anosov representations), introduced by Labourie and further developed
by Guichard-Wienhard and Kapovich-Leeb-Porti, extend the notion of convex-cocompactness
to the higher-rank. In this talk, we discuss how one can similarly understand the Hausdorff dimension
of the limit sets of Anosov subgroups in terms of their appropriate critical exponents.
This is a joint work with Michael Kapovich.
Monday October 28, 2019 at 3:00 PM in 636 SEO