Algorithms for the Tits alternative and related problems

Abstract

We present an algorithm that decides whether a finitely generated linear group over an infinite field is solvable-by-finite: a computationally effective version of the Tits alternative. We also give algorithms to decide whether the group is nilpotent-by-finite, abelian-by-finite, or central-byfinite. Our algorithms have been implemented in MAGMA and are publicly available.

Algorithms for linear groups of finite rank

“…Now let F be arbitrary and G ≤ GL(n, F) be finitely generated SF. In [7,Section 4] we show how to test whether G is completely reducible. Here we describe a more general procedure.…”

“…We refer to [7,Section 3.2]. The computations carried out in a run of IsSolvableByFinite(G) yield a change of basis matrix x such that G x is block upper triangular and all diagonal blocks of G x ρ are abelian.…”

Recognizing finite matrix groups over infinite fields

“…We test unipotency of the congruence subgroup K G in step (5) using the normal generating set K . A procedure for doing this, based on computation in enveloping algebras, is given in [11,Section 5.2]. Also note that we can apply a conjugation isomorphism as in [15] to write the SW-image over the smallest possible finite field of the chosen characteristic.…”

The constructive membership problem for discrete free subgroups of rank 2 of

“…Algorithms to decide the Tits alternative over the rational field Q have been described by Beals [2], Ostheimer [13] and Assmann and Eick [1]. An algorithm for arbitrary fields has been introduced by Detinko, Flannery and O'Brien [7].…”

“…In this case, further structural investigations of the group G are possible, see for example [1] and [7]. The case that the matrix group G contains a non-cyclic free subgroup is considered to be the 'wild' case.…”