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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.