We show that a Fuchsian group, acting on the upper half-plane model for ‫ވ‬ 2 , admits a Ford domain which is also a Dirichlet domain, for some center, if and only if it is an index 2 subgroup of a reflection group. This is used to exhibit an example of a maximal arithmetic hyperbolic reflection group which is not congruence. Analogous results, and counterexamples, are given in the case of Kleinian groups.

We establish a dimension formula for the mod 2 cohomology of finite index subgroups in the Bianchi groups (SL 2 groups over the ring of integers in an imaginary quadratic number field). For congruence subgroups in the Bianchi groups, we provide an algorithm with which the parameters in our formula can be explicitly computed. We prove our formula with an analysis of the equivariant spectral sequence, combined with torsion subcomplex reduction.arXiv:1707.06078v2 [math.KT]

For d 4, we describe an elementary construction of nonzero degree, strict contractions between closed, oriented hyperbolic d-orbifolds. Appealing to work of Guéritaud-Kassel and Tholozan, these examples determine exotic quotients of SO0(d, 1).

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