Abstract. Deligne and Mostow constructed a class of lattices in P U(2, 1) using monodromy of hypergeometric functions. Thurston reinterpreted them in terms of cone metrics on the sphere. In this spirit we construct a fundamental domain for the lattices with three fold symmetry in the list of Deligne and Mostow. This is a generalisation of the works of Boadi and Parker and gives a different interpretation of the fundamental domain constructed by Deraux, Falbel, and Paupert.
Let S be a connected non-orientable surface with negative Euler characteristic and of finite type. We describe the possible closures in $$\mathcal M\mathcal L$$
M
L
and $$\mathcal P\mathcal M\mathcal L$$
P
M
L
of the mapping class group orbits of measured laminations, projective measured laminations and points in Teichmüller space. In particular we obtain a characterization of the closure in $$\mathcal M\mathcal L$$
M
L
of the set of weighted two-sided curves.
We classify representations of a class of Deligne-Mostow lattices into PGL(3, C). In particular, we show local rigidity for the representations (of Deligne-Mostow lattices with 3-fold symmetry and of type one) where the generators we chose are in the same conjugacy class as the generators of Deligne-Mostow lattices. We use formal computations in SAGE to obtain the results. The code files are available on GitHub ([FPUP21]).
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