Abstract |
For a surface with constant Gaussian curvature, various natural
quantities grow exponentially quickly if curvature is negative,
polynomially quickly if curvature is zero, and not at all if curvature
is positive. I will describe this trichotomy for the number of
(homotopy classes of) closed geodesics; then I will explain how
Margulis proved even more precise asymptotic estimates in the case of
negative curvature, and how his results have been extended to examples
where curvature is zero or even positive in some places. Finally, I
will describe joint work with Gerhard Knieper and Khadim War in which
we obtain Margulis asymptotics for surfaces without conjugate points.
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