By C. M. Campbell, M. R. Quick, E. F. Robertson, C. M. Roney-Dougal, G. C. Smith, G. Traustason

Teams St Andrews 2009 was once held within the collage of tub in August 2009 and this moment quantity of a two-volume ebook includes chosen papers from the foreign convention. 5 major lecture classes got on the convention, and articles in keeping with their lectures shape a considerable a part of the complaints. This quantity comprises the contributions via Eammon O'Brien (Auckland), Mark Sapir (Vanderbilt) and Dan Segal (Oxford). except the most audio system, refereed survey and examine articles have been contributed through different convention contributors. prepared in alphabetical order, those articles disguise a large spectrum of contemporary staff thought. The ordinary lawsuits of teams St Andrews meetings have supplied snapshots of the country of analysis in crew thought in the course of the previous 30 years. previous volumes have had a huge impression at the improvement of team thought and it really is expected that this quantity could be both very important.

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Constructive recognition of classical groups in their natural representation. J. Symbolic Comput. 35, 195–239, 2003. [24] Peter A. Brooksbank. Fast constructive recognition of black-box unitary groups. LMS J. Comput. Math. 6, 162–197, 2003. ´ [25] Peter Brooksbank, Alice C. Niemeyer and Akos Seress. A reduction algorithm for matrix groups with an extraspecial normal subgroup. Finite Geometries, Groups and Computation, (Colorado), pp. 1–16. De Gruyter, Berlin, 2006. [26] Peter A. Brooksbank and William M.

Then the subgroup N of F2 generated by K satisﬁes the congruence extension property: that is, for every normal subgroup L ✁ N , L F ∩ N = L. Hence H = N/L embeds into G = F2 / L . 6 In fact [Ol95] contains a much stronger result for arbitrary hyperbolic groups. Proof Consider L as the set of relations of G. We need to show that the kernel of the natural map N → G is L, that is if w(K) = 1 modulo L (here w(K) is obtained from w by plugging elements of K for its letters), then w ∈ L. Consider a van Kampen diagram ∆ for the equality w(K) = 1 with minimal possible number of cells.

3 (Borisov, Sapir [BS1]) Let P n : An (Fq ) → An (Fq ) be the n-th iteration of P . Let V be the Zariski closure of P n (An ). The set of its geometric points is V (Fq ), where Fq is the algebraic closure of Fq . Then the following hold. 1. All quasi-ﬁxed points of P belong to V (Fq ). 2. Quasi-ﬁxed points of P are Zariski dense in V . In other words, suppose W ⊂ V is a proper Zariski closed subvariety of V . Then for some Q = q m there is a point (a1 , . . , an ) ∈ V (Fq ) \ W (Fq ) Sapir: Residual properties of 1-relator groups 337 such that ⎧ f1 (a1 , .