By Thomas J. Enright

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8 BS(1, 2) = a, t | tat−1 = a2 is metabelian, and representable by matrices over a commutative ring (as are all ﬁnitely generated metabelian groups), so it is residually ﬁnite by Malcev’s result [Mal]. 6 has a positive solution. Namely we are going to discuss the following. 9 (Borisov, Sapir, Spakulov´ a [BS1], [BS2], [SS]) Almost surely as n → ∞, every 1-relator group with 3 or more generators and relator of length n, is • residually ﬁnite, • a virtually residually (ﬁnite p-group) for all but ﬁnitely many primes p, • coherent (that is, all ﬁnitely generated subgroups are ﬁnitely presented).

Assume that G = X ≤ GL(d, q) is input to CompositionTree. Some of the algorithms used in constructing a composition tree for G are Monte Carlo. To verify the resulting construction, we write down a presentation for the group deﬁned by the tree and show that G satisﬁes its relations. The output of CompositionTree is: • A composition series 1 = G0 ✁ G1 ✁ G2 ✁ · · · ✁ Gm = G. • A representation Sk = Xk of Gk /Gk−1 . • Eﬀective maps τk : Gk → Sk and φk : Sk → Gk . The map τk is an epimorphism with kernel Gk−1 ; if g ∈ Sk , then φk (g) is an element of Gk satisfying τk φk (g) = g.

There exists an inﬁnite ﬁnitely generated group that is: • residually ﬁnite • torsion • all sections are residually ﬁnite • every ﬁnite section is solvable; every nilpotent ﬁnite section is Abelian. 3 The Magnus procedure In order to deal with 1-relator groups, the main tool is the procedure invented by Magnus in the 30s. Here is an example. 1 (Magnus procedure) Consider the group a, b | aba−1 b−1 aba−1 b−1 a−1 b−1 a = 1 For simplicity, we chose a relator with total exponent of a equal 0 (as in the proof of Baumslag-Pride above, the general case reduces to this).