Seminars & Colloquia Calendar

Number Theory Seminar

Hausdorff dimension of the limit sets of Anosov subgroups

Subhadip Dey (University of California, Davis)

Location:  Hill 425
Date & time: Tuesday, 28 January 2020 at 2:00PM - 3:00PM

Abstract: {\em Patterson-Sullivan measures} were introduced by Patterson (1976) and Sullivan (1979) to study the limit sets of Kleinian groups, discrete isometry groups of hyperbolic spaces. Using these measures, they showed a close relationship between the {\em critical exponent}, $$\delta(\Gamma)$$, of a Kleinian group $$\Gamma < \mathrm{Isom}(\mathbb{H}^n)$$ and the {\em Hausdorff dimension}, $$\mathrm{Hd}(\Lambda(\Gamma))$$, of the limit set $$\Lambda(\Gamma)$$ of $$\Gamma$$. The critical exponent gives a geometric measurement of the exponential growth rate of $$\Gamma$$-orbits in $$\mathbb{H}^n$$ and, on the other hand, the Hausdorff dimension measures the {\em size} of the limit set $$\Lambda(\Gamma)$$. For {\em convex-cocompact} (or, more generally, {\em geometrically finite}) Kleinian groups $$\Gamma$$, Sullivan proved that $$\delta(\Gamma) = \mathrm{Hd}(\Lambda(\Gamma))$$. {\em Anosov subgroups}, 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 my advisor Michael Kapovich.

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