From Farey symbols to generators for subgroups of finite index in integral group rings of finite groups

Abstract

The topic of this paper is the construction of a finite set of generators for a subgroup of finite index in the unit group u(ℤG) of the integral group ring of a finite group G. The present paper is a continuation of earlier research by Bass and Milnor, Jespers and Leal, and Ritter and Sehgal who constructed such generators provided that the group G does not have a non-abelian fixed-point free epimorphic image and the rational group algebra ℚG does not have simple epimorphic images that are two-by-two matrices over either the rationals, a quadratic imaginary extension of the rationals or a non-commutative division algebra. In this paper we allow simple images of the type M2(ℚ). We will do so by introducing new additional generators using Farey symbols, which are in one to one correspondence with fundamental polygons of congruence subgroups of PSL2(ℤ). Furthermore, for each simple Wedderburn component M2(ℚ) of ℚG, the new generators give a free subgroup that is embedded in M2(ℤ).

Concentration of closed geodesics in the homology of modular curves

“…Such a symbol always exists, and one can even find one which is symmetric around 1/2 [Kul91, Section 13]. Dooms-Jesper-Konolalov [DJK10] have described an algorithm for determining Farey symbols of a general level. Consider the polygon P ( 𝑝) with vertices at ∞, at the fractions of the Farey symbol, at the midpoint of the geodesic circle connecting 𝑎 𝑖 /𝑏 𝑖 and 𝑎 𝑖+1 /𝑏 𝑖+1 for i an even index, and for an odd index i at the PGL 2 (Z)-translate of 1+𝑖 √ 3 2 lying between 𝑎 𝑖 /𝑏 𝑖 and 𝑎 𝑖+1 /𝑏 𝑖+1 (for details see [Kul91, Section 2]).…”

“…Furthermore (considering below the indices modulo ), there are even indices i such that there are odd indices i such that for the remaining free indices, there is a pairing satisfying Such a symbol always exists, and one can even find one which is symmetric around [Kul91, Section 13]. Dooms–Jesper–Konolalov [DJK10] have described an algorithm for determining Farey symbols of a general level.…”

From the Poincaré Theorem to generators of the unit group of integral group rings of finite groups

“…In [9], Dooms, Jespers and Konovalov introduced a method (also by computing a fundamental polyhedron of a discrete group of finite covolume) to deal with exceptional simple components of the type M 2 (Q). New generators are introduced, using Farey symbols, which are in one to one correspondence with fundamental polygons of congruence subgroups of P SL 2 (Z).…”