Titles and abstracts​

 

Friday, May 23th (room S02)

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• 15:00 - 15:50 Matan Seidel (Tel Aviv University)

Primitivity Testing in Free Group Algebras via Duality

Let F be a free group and K a field. The free group algebra K[F] bears a strong resemblance to F, making it an excellent tool in the study of free groups. For example, by a theorem due to Cohn and Lewin, one-sided ideals in K[F] are free as K[F]-modules, analogously to the Nielsen-Schreier theorem. I will discuss this resemblance, along with other motivations for our interest in K[F] arising from the theory of word measures. I will then present a new algorithm for deciding if a given element is part of some basis of a given ideal, similarly to what Whitehead's algorithm performs in free groups. Based on joint work with Danielle Ernst-West and Doron Puder.​​​

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• 16:00 - 16:50 Claude Marion (Universidade do Porto)

On finite and profinite groups with the Magnus property

We investigate finite and profinite groups with the Magnus property (MP), where a group is said to have the Magnus property if whenever two elements generate the same normal subgroup then the elements are conjugate or inverse-conjugate. In particular we observe that a finite MP group must be solvable and  give the classification of the finite primitive MP groups. This has a couple of applications such as the fact that the Fitting height of a finite MP group is at most 2, and the characterisation of the primes dividing the order of a finite  MP group. We also describe the  chief factors of a finite MP group.

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More generally we show that a profinite MP group must be prosolvable and that any quotient of a profinite MP group is again MP. Combining the latter with our results on finite MP groups, we obtain a number of corollaries, in particular we prove that a finitely generated profinite MP group is finite. This is based on joint work with Martino Garonzi and Pavel Zalesskii.

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• 17:00 - 17:50 Juan González-Meneses (Universidad de Sevilla)

Group-theoretical computation of canonical reduction systems​

Joint work with María Cumplido and Davide Perego. Recently, some geometric and topological objects, procedures and properties of braid groups are being translated to algebraic terms, so that they can be generalized to other Artin groups of spherical type. One important example is the complex of curves, which can be described as a complex of irreducible parabolic subgroups. The canonical reduction system of a braid is a distinguished set of curves, invariant under the action of the braid, along which one performs Thurston's decomposition. We give a group-theoretical algorithm to compute this set of curves, using the Garside structure of braid groups.​​​

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Saturday, May 24th (room 101)

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• 11:00 - 11:50 Jérôme Los (Aix-Marseille Université)

Some groups from some dynamics

​I'll present recent results relating some special classes of dynamical systems, given as maps on a space, with a group acting on the same space. At the end of the game we will see some very well known groups seen from an unusual point of views. The main characters of the story is a class of discontinuous piecewise homeomorphisms of the circle and how to construct a group from such maps.​​​

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• 12:00 - 12:50 Vicent Pérez Calabuig (Universitat de València)

On Dehn’s decisions problems for skew braces

In group theory, the greatest representatives of algorithmic problems are the celebrated Dehn's decision problems: the Word Problem, the Conjugacy Problem and the Isomorphism Problem. All of them were formulated with finitely presented groups in mind. Although they are unsolvable for arbitrary finitely presented soluble groups, one of the milestones of the theory is the solvability of all three problems for the class of polycyclic-by-finite groups.

 

From a brace theoretical point of view, it is natural to set out decision problems that could help to address the major problem of classifying infinite skew braces and associated solutions of the Yang-Baxter equation (YBE). The main aim of this talk is to present analogues of the celebrated Dehn's problems for skew braces, and study its solvability in the context of providing a first exponent of an algorithmic approach to the algebra of the YBE.

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