Next upcoming talk:

Recently, work of Long and Thistlethwaite, Weir, and Alessandrini-Lee-Schaffhauser generalized some of the theory of higher Teichmüller spaces to the setting of orbifold surfaces. In particular, they compute the dimension of Hitchin components for triangle groups, and find that this dimension is 0 only for a finite number of low-dimensional examples. In contrast with these results and with the torsion-free surface group case, we show that the composition of the geometric representation of a hyperbolic triangle group with the diagonal embedding in PGL(2n,R) or PSp(2n,R) is always locally rigid.

Past talks:

In this talk, we will describe some fundamental domains for subgroups of PU(2,1) in the complex hyperbolic plane and in its boundary at infinity, and we will focus on the examples constructed by Parker-Will and Deraux-Falbel. We will recall some geometric properties of the complex hyperbolic plane, as well as the hypersurfaces called bisectors. The fundamental domains that we consider can be used for proving the discreteness of some groups, and provide geometric structures on manifolds.

We give effective estimates for the number of saddle connections on a translation surface that have length less or equal than L and are in a prescribed homology class modulo q. Our estimates apply to almost all translation surfaces in a stratum of the moduli space of translation surfaces, with respect to the Masur-Veech measure on the stratum. This is joint work with Michael Magee, and builds on recent advances by Magee on Selberg's eigenvalue conjecture for moduli spaces of abelian differentials, and Rodolfo Gutierrez-Romo's work on Zorich's conjecture on Zariski-density of Rauzy-Veech groups.

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