Notes of Ursula Hamenstaedt’s Cambridge lecture 09-03-2017

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<b> Surface subgroups in lattices in classical simple Lie groups </b>

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<blockquote><b>Theorem 1 (Hamenstaedt, Kahn-Labourie-Mozes)</b> <em> Let ${\Gamma be a cocompact lattice. Then ${\Gamma}$ contains surface subgroups. </em></blockquote>

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The theorem applies to all classical simple Lie groups except a few low dimensional exceptions like ${SO(2,2)}$. It probably works for irreducible lattices in products, but I have not yet checked this. Kahn-Labourie-Mozes have obtained this result independently. In fact, their result is a bit stronger, they assert that the representations of surface groups they construct are Anosov, hence have a circle as limit set in the Furstenberg boundary, and these limit sets are dense.
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Gromov raised the question of existence of surface subgroups in one-ended hyperbolic groups long ago. Calegari-Walker showed that this is indeed true for random groups. Kahn-Markovic showed that this is true for cocompact lattices in ${PSl(2,{\mathbb C})}$. They prove more: any two points in the boundary sphere are separated by the limit set of such a quasi-convex subgroup. I extended this to cocompact lattices in rank one simple groups, except possibly ${SO(2n,1)}$, a case that we completed later with Kahn.
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<b>1. Higher rank </b>

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Let ${N}$ be the symmetric space of ${SO(2,3)}$. The geodesic flow preserves direction within Weyl chambers. In particular, the barycentric direction (in which the volume growth is maximal) is invariant, whence the barycenter flow.
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Let ${\gamma}$ be a closed regular geodesic in ${M=\Gamma\setminus N}$. Then ${\gamma}$ is contained in a compact flat 2-torus (one can use arithmeticity to prove this, but one need not). One can count such regular tori according either to their systoles ${\ell(T)}$ or to their volumes ${vol(T)}$.
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<blockquote><b>Theorem 2 (Deitmar)</b> <em> Let ${h>0}$ be the exponent of growth of volume of balls. Then <p align=center>$\displaystyle \begin{array}{rcl} e^{-hR}\left(\sum_{\ell(T)\leq R}\lambda_T\right)\rightarrow \mu \quad \textrm{as}\quad R\rightarrow\infty, \end{array}$</p>
where ${\lambda_T}$ is the Liouville measure on ${T^1 T}$ and ${\mu}$ is the Liouville measure on ${T^1 M}$. </em></blockquote>

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Every regular geodesic turns out to be contained in a totally geodesic embedded ${H^3}$ (orbit of ${SO(3,1)\subset SO(3,2)}$), but they need not close up.
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<b>2. Pants gluing </b>

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The idea goes back to Kahn-Markovic: obtain closed surface by gluing pants. We use the dynamics of the barycenter flow to produce closed geodesics from almost closed ones. Also, a frame bundle version of barycenter flow is used.
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I do not explain how the pairs of pants are obtained from the totally geodesic embedded ${H^3}$‘s, it is very similar to Kahn-Markovic. Take two pairs if pants which are almost flip-symmetric around a common closed geodesic ${\gamma}$. Using exponential mixing, one arrange them to be close, and glues them into surfaces. The hardest step is to ensure they are incompressible. Kahn-Markovic require angles at which they meet to be very small, this would be very hard to adapt to higher rank. Instead, we exploit flat tori containing closed geodesics. We cut surfaces open again a closed geodesic ${\gamma}$, and insert a long flat annuli from the torus. This amounts to conjugating a pant group by the stabilizer of the torus containing ${\gamma}$. This creates thick portions in the surface, which become quasi-convex.
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