Unique product groups and congruence subgroups

Congruence Subgroups of Groups Commensurable with PSL

One-Cusped Congruence Subgroups of Bianchi Groups

We show that there are only finitely many maximal congruence subgroups of the Bianchi groups such that the quotient by H has only one cusp.

Constructing Non-congruence Subgroups of Flexible Hyperbolic 3-manifold Groups

We give an explicit construction for non-congruence subgroups in the fundamental group of a flexible hyperbolic 3-manifold.

Representations of Braid Groups via Conjugation Actions on Congruence Subgroups

where α and β are nonzero complex numbers. If α, β lie in some subfield F of C, then the representations are defined over F (in fact, over Z[α±1, β±1]). The starting point for ρn(α) is the reduced Burau representation βn : Bn → GLn−1(Z[t, t −1]). For each i ≥ 1, set K(α) = {A ∈ SLn−1(C[t, t ]) : A ≡ I mod (t− α)}. The sequence {K(α)}i≥1 is a central series inK(α) = K 1(α) (i.e., Ki+j(α) ⊇ [K(α)...

Algebraic Curves Uniformized by Congruence Subgroups of Triangle Groups

We construct certain subgroups of hyperbolic triangle groups which we call “congruence” subgroups. These groups include the classical congruence subgroups of SL2(Z), Hecke triangle groups, and 19 families of arithmetic triangle groups associated to Shimura curves. We determine the field of moduli of the curves associated to these groups and thereby realize the groups PSL2(Fq) and PGL2(Fq) regul...