Tim Hsu

Identifying congruence subgroups of the modular group

We exhibit a simple test (Theorem 2.4) for determining if a given (classical) modular subgroup is a congruence subgroup, and give a detailed description of its implementation (Theorem 3.1). In an appendix, we also describe a more “invariant” and arithmetic congruence test. 1. Notation We describe (conjugacy classes of) subgroups Γ ⊂ PSL2(Z) in terms of permutation representations of PSL2(Z), following Millington [11, 12] and Atkin and Swinnerton-Dyer [1]. We recall that a conjugacy class of subgroups of PSL2(Z) is equivalent to a transitive permutation represention of PSL2(Z). Such a representation can be defined by transitive permutations E and V which satisfy the relations 1 = E = V . (1.1) The relations (1.1) are fulfilled by E = ( 0 1 −1 0 ) , V = ( 1 1 −1 0 ) . (1.2) Alternately, such a representation can be defined by transitive permutations L and R which satisfy 1 = (LR−1L)2 = (R−1L)3, (1.3) with the relations being fulfilled by L = ( 1 1 0 1 ) , R = ( 1 0 1 1 ) . (1.4) One can also use permuations E and L such that 1 = E = (L−1E)3, (1.5) with E and L corresponding to the indicated matrices in (1.2) and (1.4), respectively. Received by the editors September 1, 1994. 1991 Mathematics Subject Classification. Primary 20H05; Secondary 20F05.

Cubulating graphs of free groups with cyclic edge groups

We prove that if

Partitioning the Boolean Lattice into Chains of Large Minimum Size

G

Ascending HNN extensions of polycyclic groups are residually finite

is a group that splits as a finite graph of finitely generated free groups with cyclic edge groups, and

Partitioning the boolean lattice into a minimal number of chains of relatively uniform size

G

Explicit constructions of code loops as centrally twisted products

has no non-Euclidean Baumslag-Solitar subgroups, then

Artin HNN-extensions virtually embed in Artin groups

G

Embedding theorems for non-positively curved polygons of finite groups

is the fundamental group of a compact nonpositively curved cube complex. In addition, if

Groups with infinitely many types of fixed subgroups

G

Methods for nesting rank 3 normalized matching rank-unimodal posets

is also word-hyperbolic (i.e., if