Past Algebra Seminar

8 February 2011
17:00
Dr Ehud Meir
Abstract
Abstract: this is joint work with Eli Aljadeff. Let G be a group, H a finite index subgroup. Moore's conjecture says that under a certain condition on G and H (which we call the Moore's condition), a G-module M which is projective over H is projective over G. In other words- if we know that a module is ``almost projective'', then it is projective. In this talk we will survey cases in which the conjecture is known to be true. This includes the case in which the group G is finite and the case in which the group G has finite cohomological dimension. As a generalization of these two cases, we shall present Kropholler's hierarchy LHF, and discuss the conjecture for groups in this hierarchy. In the case of finite groups and in the case of finite cohomological dimension groups, the conjecture is proved by the same finiteness argument. This argument is straightforward in the finite cohomological dimension case, and is a result of a theorem of Serre in case the group is finite. We will show that inside Kropholler's hierarchy the conjecture holds even though this finiteness condition might fail to hold. We will also discuss some other cases in which the conjecture is known to be true (e.g. Thompson's group F).
30 November 2010
17:00
Tatiana Bandman
Abstract
I will speak about a geometric method, based on the classical trace map, for investigating word maps on groups PSL(2, q) and SL(2, q). In two different papers (with F. Grunewald, B. Kunyavskii, and Sh. Garion, F. Grunewald, respectively) this approach was applied to the following problems. 1. Description of Engel-like sequences of words in two variables which characterize finite solvable groups. The original problem was reformulated in the language of verbal dynamical systems on SL(2). This allowed us to explain the mechanism of the proofs for known sequences and to obtain a method for producing more sequences of the same nature. 2. Investigation of the surjectivity of the word map defined by the n-th Engel word [[[X, Y ], Y ], . . . , Y ] on the groups PSL(2, q) and SL(2, q). Proven was that for SL(2, q), this map is surjective onto the subset SL(2, q) $\setminus$ {−id} $\subset$ SL(2, q) provided that q $\ge q_0(n)$ is sufficiently large. If $n\le 4$ then the map was proven to be surjective for all PSL(2, q).

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