**PRELECTURA DE TESIS**

**Doctorando: Diego López Álvarez.**

Título: Sylvester rank functions, epic division rings and the strong Atiyah conjecture for locally indicable groups.

Director de tesis: Andrei Jaikin.

Fecha y Hora: 2 de diciembre, 12:00.

Lugar: ONLINE-mediante una reunión virtual en el equipo de Microsoft Teams titulado “Prelectura tesis Julio Aroca/Diego López (2 diciembre)

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Abstract:

Throughout the thesis we consider questions related to embeddings of noncommutative domains into division rings.

We first treat the problem of existence of such an embedding for group rings K[G] where K is a subfield of the field of complex numbers C and G is a locally indicable group (for instance, a torsion-free one-relator group).

In this sense, the strong Atiyah conjecture for these group rings, principal and original motivation to the thesis, proposes a candidate to be a division ring in which K[G] embeds, namely, the division closure of K[G] inside a certain ring of operators U(G) related to the group von Neumann algebra N(G). In the main result (joint with A. Jaikin-Zapirain) we prove that the strong Atiyah conjecture holds in this setting and that, moreover, the resulting division ring can be uniquely identified through a universal property. The associated methods and results allow us to prove a posteriori other related conjectures, such as a version of Lück's approximation conjecture for virtually locally indicable groups.

In the second place, we deal with the notion of universality of a division ring. For a ring R, a universal division ring of fractions is a division ring that contains and is generated by R as a division ring, and in which we can invert ``the most'' matrices possible over $R$. In this regard, (pseudo)-Sylvester domains are rings R admitting a universal division ring of fractions in which every matrix becomes invertible unless there is an ``obvious'' obstruction.

In a joint work with F. Henneke, we prove that crossed products of the form E*G, where E is a division ring and G is free-by-{infinite cyclic}, are always pseudo-Sylvester domains, and we explore the more general situation of crossed products F*Z of a fir F and the ring of integers Z.

Along the thesis, the theory of Sylvester rank functions provides a unifying language and a tool to address the problems considered. We further analyze the space of Sylvester rank functions that can be defined on certain families of rings, including Dedekind domains and a subfamily of skew Laurent polynomial rings with coefficients in a division ring (joint with A. Jaikin-Zapirain).

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**Doctorando: Julio Aroca Lobato.**

Título: On Dehn's decision problems in generalized Thompson's groups.

Director de tesis: Javier Aramayona.

Tutor de tesis: Andrei Jaikin.

Fecha y Hora: 2 de diciembre, 10:30.

Lugar: ONLINE-mediante una reunión virtual en el equipo de Microsoft Teams titulado “Prelectura tesis Julio Aroca/Diego López (2 diciembre)

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Abstract:

Introduced by Richard Thompson in 1965, Thompson's classical groups F, T and V are important examples of finitely presented subgroups of the group of homeomorphisms of the Cantor set. Historically, T and V were the first examples of infinite, finitely presented simple groups. The main focus of this thesis is a family of generalizations of Thompson's groups, called symmetric Thompson's groups, originally introduced by Hughes in the context of Finite Similarity Structure (FSS) groups. They are subgroups of the homeomorphism group of the Cantor set, but now the "local action" is governed by a fixed subgroup of an appropriate symmetric group.

Our first result is a solution of the conjugacy problem for symmetric Thompson's groups. More concretely, we generalize a construction of Belk-Matucci in order to interpret conjugacy in these groups in terms of certain topological objects called strand diagrams. As a consequence, we are able to conclude that a large class of symmetric Thompson's groups are pairwise non-isomorphic. Next, in a joint work with C. Bleak (St. Andrews), we give conditions that guarantee the existence of an embedding between pairs of symmetric Thompson's groups, extending results of Higman and Birget. Moving away from symmetric Thompson's groups, in collaboration with M. Cumplido (UCM) we introduce a new family of groups that generalize the so-called braided Thompson's groups of Brin and Dehornoy. Roughly speaking, the difference between our groups and those of Brin and Dehornoy is that we allow for infinite, but recursive, braiding. We show that, as is the case with braided Thompson's groups, every infinitely braided Thompson's group is finitely generated.

Thompson's groups are also strongly related to mapping class groups of infinite-type surfaces, through the so-called asymptotic mapping class groups, which have been extensively studied by Sergiescu, Funar and Kapoudjian mainly. Our final result is a rigidity result for a simplicial complex defined in terms of simple closed curves on a surface; more concretely, we prove that its automorphism group is the extended mapping class group, extending a result of Margalit to the infinite-type setting.