Groups and geometry

Sebastian Bischof: Groups of Kac-Moody type over $F_2$
Twin buildings are combinatorial objects which were introduced by Ronan and Tits in the late 1980s. Their definition was motivated by the theory of Kac-Moody groups over fields and they are natural generalizations of spherical buildings. Spherical buildings were classified by Tits in the 1970s and this result is based on a local-to-global result on spherical buildings. Tits asked the question whether a similar result holds for twin buildings. This has been confirmed by Mühlherr and Ronan under an additional assumption which is not satisfied for twin buildings associated with Kac-Moody groups over $F_2$. We give a construction of groups of Kac-Moody type over $F_2$ which shows that the local-to-global result does not hold in general. We will discuss some applications including finiteness properties and Property $T$.

Sira Busch: Projectivity Groups of Spherical Buildings
Buildings are combinatorial and geometric structures that were introduced by Jacques Tits for the study of semi-simple algebraic groups. When a building has the so-called Moufang property, it corresponds to the building having a rich automorphism group. An interesting subgroup of the automorphism group of a Moufang-building is the little projective group, which is the group generated by all root elations. In joined work with Hendrik Van Maldeghem, we found geometrical constructions for root elations of Moufang-buildings of finite diameter that give more insight into their fixpoint structure. We also saw a connection between the little projective groups and the groups of special projectivities of these buildings. At the moment we are working on determining the special and general groups of projectivities for all Moufang-buildings of finite diameter together with Jeroen Schillewaert. In my talk I would like to give some insight into our research, explain what elations and projectivities are and state some group theoretic consequences.

Ilaria Castellano: Totally disconnected locally compact groups, accessibility and Euler-Poincaré characteristic
In the first part of the talk I will illustrate how the classical notion of accessibility for finitely generated groups carries over to the realm of compactly generated totally disconnected locally compact (= t.d.l.c.) groups. Then, by means of a new notion of Euler-Poincaré characteristic, I will discuss an accessibility result in the t.d.l.c. framework, under the assumption of rational discrete cohomological dimension $= 1$.

Maria Cumplido: Retractions to parabolic subgroups in Artin groups beyond the even case
An Artin group is defined with a presentation with a finite set of generators $S$ and, for each pair of generators $s,t \in S$, a single relation (or none) having the form $sts\cdots = tst\cdots$ where the number of letters on each side of the equality is the same and is denoted by $m_{s,t}$. Despite their simple definition, little is generally known about Artin groups (for example, the general solution to any Dehn problem is unknown). When the length $m_{s,t}$ of all existing relations is even, the Artin group is said to be even. Also, given an Artin group with its standard presentation, a standard parabolic subgroup is the subgroup generated by a subset of generators. In even Artin groups, there exists an obvious retraction to their standard parabolic subgroups that sends a generator to itself if it belongs to the subgroup and to 1 otherwise. Recently, Antolín and Foniqi showed that these retractions are immensely useful for obtaining new results in even Artin groups. This joint work with Bruno A. Cisneros de la Cruz and Islam Foniqi aims to answer the natural question: what happens when we introduce odd-length relations? How can we define retractions, and how do they generalize the existing results?

Dawid Kielak: One-relator groups
I will survey the history of the subject, and explain a number of recent developments.

Jon McCammond: An Artin Group Family Reunion
Back in the 1970s, when Artin groups were initially defined and studied, they were closely associated with singularity theory and the Arnold school. There is, for example, a close connection between isolated singularities and spherical type Artin groups. In this talk, I will try to connect the study of general Artin groups to this broader context and to a number of objects which can be thought of as relatives of Artin groups.

Ismael Morales: $L^2$-homology and fixed points of automorphisms of free groups
Many useful techniques in group theory have been originally developed for the purpose of studying problems regarding subgroups of free groups and their ranks. An illustrative example is a conjecture attributed to Scott, solved by Bestvina and Handel, which bounds the rank of the subgroup fixed by any automorphism. We will discuss a new approach to this problem based on $L^2$-homology.

Bernhard M. Mühlherr: Tits Polygons
Tits polygons are generalizations of Moufang polygons. Examples of Tits polygons arise from spherical Moufang buildings by a procedure that involves a Tits index. It is an open question whether each Tits polygon can be obtained in this way. In my talk I will introduce Tits polygons, discuss their motivation and report on our ongoing efforts to answer the above question. This is joint work with Richard Weiss.

Luis Paris: Artin groups of type $D_n$
Recall that an Artin group is called of spherical type if its associated Coxeter group is finite. It is known that the irreducible spherical Artin groups consist of four infinite families, the groups of type $A_n$, the groups of type $B_n$, the groups of type $D_n$ and the groups of type $I_2 (m)$, and $6$ sporadic groups. The groups of type $A_n$ are the braid groups and they can be easily understood using various algebraic and/or topological methods. The groups of type $B_n$ are finite index subgroups of the groups of type $A_n$, hence the methods used to understand the groups of type $A_n$ often apply in this case as well. The groups of type $I_2 (m)$ are virtually direct products of a free group with $\mathbb Z$. However, in order to understand the groups of type $D_n$, it is often necessary to develop new tools and they are more difficult to study. This talk will recount the different studies on these groups to culminate in the latest one, in collaboration with Fabrice Castel, where we classify the endomorphisms of such groups for a large $n$.

Bertrand Remy: Wave equations on affine buildings and application to entropy lower bounds
This talk will deal with (harmonic) analysis of affine buildings. We will discuss the construction of kernels associated with discrete multitemporal wave equations on these spaces. One motivation is to contribute to progress in arithmetic quantum unique ergodicity, following a general strategy due to Brooks and Lindenstrauss.
Thias is joint work with Jean-Philippe Anker and Bartosz Trojan.

Sarah Rees: Compressed decision problems
I'll be talking about joint work with Derek Holt, which examines the compressed word and conjugacy problems, specifically for relatively hyperbolic groups. Rather than focus on the technicalities of our proof I intend to present a non-technical survey of these problems and related problems, and of the solutions to them that preceded our work, and on which our work is based.
For a group $G$ with finite generating set $X$, a solution to the word problem provides an algorithm that given a word $w$ (finite string of elements from $X \cup X^{-1}$) can determine whether or not $w$ represents the identity element of $G$, while a solution to the conjugacy problem can determine whether or not two words $u,v$ represent elements of $G$ that are conjugate within the group (i.e. $gu=_G vg$ for some $g \in G$.
However, in some situations an element of $G$ is not most naturally represented as a string over $X$ but rather by a sequence of rules from a grammar through which it is produced, e.g. the string $aaaaaaaabbbbcc\,(=a^8b^4c^2)$ can be constructed as a concatenation of three powers, each produced by repeated squaring of a generator ($a$, $b$ or $c$); such a representation is called a compressed word. Solution to the compressed word and conjugacy problems determine whether a compressed word represents the identity element, or whether two compressed words represent conjugate elements.
Compressed word descriptions of group elements can arise very naturally; we observe that Schleimer's 2008 result that the word problem for $Aut(F_m)$ can be solved in polynomial time is a consequence of Lohrey's 2006 result that the compressed word problem of $F_m$ has a polynomial time solution.
We shall see how solutions for the compressed word problems for first free, then hyperbolic, and then relatively hyperbolic groups all make use of the underlying geometry of the Cayley graph(s) over the groups, over one or more generating sets.

Damiano Rossi: The Brown complex of a finite reductive group
Let $G$ be a finite group and $p$ a prime. The Brown complex of $G$ at the prime $p$ is the simplicial complex associated with the poset of non-trivial $p$-subgroups of $G$. The topology of this complex plays a crucial role in understanding the algebraic $p$-local structure of the group and has significant implications for related representation theoretic questions. Quillen has shown that the Brown complex of a finite reductive group defined over a field of characteristic $p$ is homotopy equivalent to the Tits building. In this talk, we consider the case of a finite reductive group defined over a field of characteristic different from $p$ and show that the homotopy type of the Brown complex can be described in terms of the generic Sylow theory introduced by Broué and Malle.

Mireille Soergel: Dyer groups: Coxeter groups, right-angled Artin groups and more...
Dyer groups are a family encompassing both Coxeter groups and right-angled Artin groups. Among many common properties, these two families admit the same solution to the word problem. Each of these two classes of groups also have natural piecewise Euclidean CAT(0) spaces associated to them. In this talk I will introduce Dyer groups, give some of their properties.

Stefan Witzel: Arithmetic groups and their relatives
Arithmetic groups are lattices on products of buildings and symmetric spaces. With appropriate assumptions the converse is true as well. I will discuss some of these assumptions and how their failure can lead to interesting groups.