Research Seminar in Discrete Mathematics and Algebra (DMA)

Current semester:

• 10.7.2024 Various speakers: Workshop on sofic graphs.
  
  
• 3.7.2024 Will J. Turner: Local Separators of Cayley Graphs: Proof Strategy.
  Abstract: Stallings' Theorem (1971) states that a finitely-generated group splits over a finite subgroup if its Cayley graphs have more than one end. In 2010, Bernhard Krön gave a short proof of Stallings' Theorem using separators of graphs. In this project, we move towards a finite version of Stallings' Theorem using local separators. We show that a finitely-generated group (with bounded nilpotency), having a certain local separator in one of its Cayley graphs, is necessarily cyclic, dihedral or decomposes as a direct product of a cycle and an involution. In particular, we assume the existence of a d-local cutvertex or a d-local 2-separator, for d bounded below in terms of the group's nilpotency. A r-local cutvertex u in a graph is a vertex which disconnects the subgraph induced on a ball of diameter d centred at u. A d-local 2-separator is defined so that it works in a similarly intuitive way.
  
  
• 5.6.2024 Paula Kahl: Introduction to amenability and related properties of groups.
  Abstract: Introduced by Gromov in the late 1990s, the notion of a sofic group turned out to be very fruitful. Within a short period of time several long-standing group-theoretic conjectures, such as Gottschalk's surjunctivity conjecture or Kaplansky's direct finiteness conjecture (both still open for arbitrary groups), were proved for sofic groups. This suggests to think of soficity as a strong property only few groups satisfy. However, until now, there is not a single group known to be non-sofic. Starting from the older and stronger notion of amenability, the talk provides an introduction to soficity, hyperlinearity and the Haagerup property, as well as the relations between them. In particular we are going to prove that amenability implies soficity.
  
  
• 22.5.2024 Will J. Turner: Local Separators of Cayley Graphs: Towards a Stallings-Type Theorem for Finite Groups.
  Abstract: Stallings' Theorem (1971) states that a finitely-generated group splits over a finite subgroup if its Cayley graphs have more than one end. In 2010, Bernhard Krön gave a short proof of Stallings' Theorem using separators of graphs. In this project, we move towards a finite version of Stallings' Theorem using local separators. We show that a finitely-generated group (with bounded nilpotency), having a certain local separator in one of its Cayley graphs, is necessarily cyclic, dihedral or decomposes as a direct product of a cycle and an involution. In particular, we assume the existence of a d-local cutvertex or a d-local 2-separator, for d bounded below in terms of the group's nilpotency. A r-local cutvertex u in a graph is a vertex which disconnects the subgraph induced on a ball of diameter d centred at u. A d-local 2-separator is defined so that it works in a similarly intuitive way.
  
  
• 15.5.2024 Tim Planken: Colouring d-dimensional triangulations.
  Abstract: We consider d-dimensional triangulations and show structural equivalences for (d+1)-colourability and (d+2)-colourability of the vertices of such triangulations. In particular, we show that a d-dimensional triangulation admits a proper (d+1)-colouring if and only if every (d-2)-cell is incident with an even number of (d-1)-cells. Moreover, a d-dimensional triangulation C admits a proper (d+2)-colouring if and only if there exists a triangulation C' that contains C such that for every (d-2)-cell in C', the number of incident (d-1)-cells is divisible by three.
  
  
• 8.5.2024 Josefin Bernhard: Automatic continuity of topological groups.
  Abstract: Many concrete groups come equipped with a non-trivial compatible topology. In some natural examples of topological groups, the topology is determined already by the algebraic structure. Such situation is closely linked with the automatic continuity property: a topological group is said to have automatic continuity if every homomorphism from it into any separable topological group is continuous. In the talk, examples and applications of this phenomenon will be given and we will discuss methods of proving this property for a given group.
  
  
• 29.4.2024 Arkady Leiderman (Ben-Gurion, Israel): Embedding the free topological group F(X^n) into F(X).
  
  
  

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