Algorithms for linear groups of finite rank

Abstract

Let G be a finitely generated solvable-by-finite linear group. We present an algorithm to compute the torsion-free rank of G and a bound on the Prüfer rank of G. This yields in turn an algorithm to decide whether a finitely generated subgroup of G has finite index. The algorithms are implemented in MAGMA for groups over algebraic number fields.

Homological dimension of elementary amenable groups

“…Proof. Suppose first that G is a finitely generated polyminimax group and that H is a subgroup of Hirsch length n. A result of Robinson which uses the fact that G is virtually solvable with finite abelian ranks (see [8,Theorem 3.1]) shows that H has finite index in G in this case.…”

Practical Computation with Linear Groups Over Infinite Domains

“…More generally, these algorithms work for solvable-by-finite groups G over any field, albeit with qualifications on G in positive characteristic. The papers [18,19] contain lengthier discussion of the above.…”

“…Implementations are available in Magma; see [17]. Experimental results are reported in [18,Section 6], [19,Section 4.5], and [20, Section 5].…”

Linear groups and computation

“…First, a presentation of CRPart(S) is employed to produce a set Y of normal generators for U (G). Secondly, a generating set of H as in Proposition 4.15 is found by the method in [44,Section 4.3]; RankU is required and the computation initializes at Y . Then h(U (G)) = h(H).…”