Abstract
We study free subgroups of index four of the parametrized modular group Π, the subgroup of SL(2; Z[ξ]) generated by (10 ξ1) and (01 -10). There are eight free subgroups, four of which are normal and four are non-normal. Then we study the intersections of the normal subgroups. We give canonical presentations in terms of generators and relations. At the end of the paper we study connections between and the Bianchi groups, the two-parabolic group and a group from relativity theory.
Highlights
The parametrized modular group Π is defined in [10] as the subgroup of SL(2, Z[ξ]) generated by 0 −1 A=, B= (1)where Z[ξ] is the polynomial ring over Z with ξ as indeterminate
At the end of the paper we study connections between Π and the Bianchi groups, the two-parabolic group and a group from relativity theory
In the last section we describe some connections with the Picard group and other Bianchi groups using the results of R
Summary
The parametrized modular group Π is defined in [10] as the subgroup of SL(2, Z[ξ]) generated by. The previous paper [10] studied analytical properties of the singular set of Π and the enumeration of the elements of Π, see Lemma 2.1 below. The present paper investigates Π more in the spirit of combinatorial group theory [9] [7]. The exponent sums of a word W ∈ Π with respect to the generators (1) are σ(W ) := (sum of exponents of A in W ),. Which defines a homomorphism of Π into the additive group Z, and τ (W ) := (sum modulo 4 of exponents of B in W ),. Which defines a homomorphism of Π into the additive group Z/4Z, note that B4 = I
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