Infinite Group Theory 2015

s Prof. Paul Baginski Title: Definability and Nilpotence in Groups with Bounded Chains of Centralizers Abstract: A group G has bounded chains of centralizers (G is MC) if every chain of centralizers CG(A1) ≤ CG(A2) ≤ . . . is finite. While this class of groups is interesting in its own right, within the field of logic known as model theory, MC groups have been examined because they strictly contain the class of stable groups. Stable groups, which have a rich literature in model theory, robustly extend ideas such as dimension and independence from algebraic groups to a wider setting, including free groups. Stable groups gain much of their strength through a chain condition known as the Baldwin-Saxl chain condition, which implies the MC property as a special case. Several basic, but key, properties of stable groups have been observed by Wagner and others to follow purely from the MC condition (this builds upon the work of Bludov, Khukhro, and others). These properties were, as one would expect, purely group-theoretic. From the perspective of logic, the class of MC groups should be unruly, since the MC cannot be captured using first-order axioms (unless one insists on a fixed finite centralizer dimension). Yet we have uncovered that MC groups possess a logical property of stable groups as well, namely the abundance of definable nilpotent subgroups. We shall present this result and describe the current investigations for finding an analogue for solvable subgroups. Joint work with Tuna Altınel. Prof. Volker Diekert Title: Amenability and Conjugacy Abstract: The talk is based on a joint work with Alexei Myasnikov and Armin Weiß. In a paper presented at LATIN 2104 we investigated the conjugacy problem in Baumslag’s group BG(1, 2) = 〈a, b | (bab−1)a(bab−1)−1 = a〉; and we proved that its conjugacy problem BG1,2 is decidable in polynomial time in a strongly generic setting. In the present talk we take a more general viewpoint. We examine Schreier graphs of amalgamated products and HNN extensions. For an amalgamated product G = H ?A Kwith [H : A] ≥ [K : A] ≥ 2, the Schreier graph with respect to H or K turns out to be non-amenable if and only if [H : A] ≥ 3. Moreover, for an HNN extension of the form G = GenH, bbab−1 = φ(a), a ∈ A, we show that the Schreier graph of G with respect to the subgroup H is non-amenable if and only if A 6= H 6= φ(A). As application of these characterizations we show that under certain conditions the conjugacy problem in fundamental groups of finite graphs of groups with free abelian vertex groups can be solved in polynomial time on a strongly generic set. Furthermore, the conjugacy problem in groups with more than one end can be solved with a strongly generic algorithm which has essentially the same time complexity as the word problem. These are rather striking results as the word problem might be easy, but the conjugacy problem might be even undecidable. Finally, our results yield another proof that the set where the conjugacy problem of the Baumslag’s group BG(1, 2) is decidable in polynomial time is also strongly generic. The results are from a paper which will appear at ISSAC 2015. Prof. Ben Fine Title: On CT and CSA Groups