The Buildings conference is an annual meeting and in 2024 we will celebrate its 30th edition. The conference will be held in honor of Linus Kramer's 60th birthday. Some of the main topics of the conference are: generalized polygons and related geometries, buildings of higher rank, geometries and groups of Lie-type, Tits magic square, algebraic varieties related to incidence geometries, automorphism groups of incidence geometries, and geometric group actions, algebraic groups and Kac-Moody groups, topological aspects of buildings and related geometries.

Speakers

Barbara Baumeister
Ruth Charney
Corina Ciobotaru
Tom De Medts
Amandine Escalier
Ralf Köhl
Norbert Knarr
Alexander Lytchak
Bernhard Mühlherr
Damian Osajda
Petra Schwer
Jeroen Schillewaert
Markus Stroppel
Maneesh Thakur
Anne Thomas
Hendrik Van Maldeghem
Olga Varghese
Richard Weiss
Stefan Witzel

Organisers

Sira Busch,
Corina Ciobotaru,

Siegfried Echterhoff,

Petra Schwer,

Olga Varghese

Conference Photo

Schedule

The following is our preliminary schedule:

Time

Monday, 23.09.24

Tuesday, 24.09.24

Wednesday, 25.09.24

Thursday, 26.09.24

09:00-09:30

Registration

09:30-10:20

Hendrik Van Maldeghem

Corina Ciobotaru

Anne Thomas

Stefan Witzel

10:20-10:30

Break

Break

Break

Break

10:30-11:20

Markus Stroppel

Norbert Knarr

Richard Weiss

Alexander Lytchak

11:20-12:00

Coffee Break

Coffee Break

Coffee Break

Coffee Break

12:00-12:50

Barbara Baumeister

Olga Varghese

Petra Schwer

Damian Osajda

12:50-14:30

Lunch Break

Lunch Break

Lunch Break

End of conference

14:30-15:20

Amandine Escalier

Ralf Köhl

Ruth Charney (Colloquium)

15:20-16:00

Coffee Break

Coffee Break

Coffee Break

16:00-16:50

Bernhard Mühlherr

Maneesh Thakur

16:50-17:00

Break

Break

17:00-17:50

Tom De Medts

Jeroen Schillewaert

18:00-...

Reception / Poster session

19:00-...

Conference dinner

Abstracts

Barbara Baumeister Extended Weyl groups
Abstract: Extended Weyl groups have been studied by several people such as Brieskorn, Saito and van der Lek. They appear at different places, for instance in the study of simple elliptic or hyperbolic singularities or in the theory of hereditary categories in representation theory of finite dimensional algebras. I will introduce extended Weyl groups, discuss their structure and some properties. Finally an application of the results will be given.

Ruth Charney Exploring the Mysteries of Artin groups
Abstract: Coxeter groups arise in many areas of mathematics including topology, algebraic geometry, representation theory, and combinatorics. They are much studied and generally well-understood. Closely related to Coxeter groups are a class of groups known as Artin groups. The best known examples of these groups are the braid groups. While easy to define, Artin groups are much more mysterious than Coxeter groups with many key properties unknown and basic questions unanswered. In this talk, I will introduce Artin groups, discuss some of the open problems, and survey some recent work, using both combinatorial and geometric methods, to approach these problems.

Corina Ciobotaru Applications of strongly I-regular hyperbolic elements to affine buildings
Abstract: For a locally finite thick Euclidean buildings X of finite Coxeter system (W, S) and for any subset I of S we define the notion of a strongly I-regular hyperbolic automorphism of X. When I is the empty set one gets the already known notion of strongly regular elements. Although the dynamics of strongly I-regular elements on the spherical building of X is much more complicated than for the strongly regulars ones, we get interesting applications. Namely, we show that for closed groups G with a type-preserving and strongly transitive action by automorphisms on X, the Chabauty limits of certain closed subgroups of G contain as a normal subgroup the entire unipotent radical of concrete parabolic subgroups of G.

Tom De Medts G_{2}- and F_{4}-graded Lie algebras and cubic norm structures
Abstract: An element x in a Lie algebra L is called extremal if [x, [x, y]] is a scalar multiple of x, for all y in L. The extremal elements give rise to an extremal geometry, and if the Lie algebra is a simple Lie algebra of an algebraic group, then typically, this extremal geometry is the shadow space of a building. We show that, under some natural conditions on the extremal geometry, the Lie algebra admits a G_{2}-grading and can be parametrized by a so-called cubic norm structure. Conversely, from any cubic norm structure, we can construct a G_{2}-graded Lie algebra. Moreover, we believe that, for a certain class of cubic norm structures known as Freudenthal algebras, the G_{2}-grading can be further refined to an F_{4}-grading. This is based on joint work with Jeroen Meulewaeter and with Torben Wiedemann.

Amandine Escalier Commensurability, measure and orbit equivalence of graph products
Abstract: Graph products form a family of groups generalising both right angled Artin groups and right angled Coxeter groups. In this talk we will study the behaviour of graph products from the point of view of measured dynamics. We will also talk about their classification up to commensurability and highlight the links with geometric group theory. If times permit we will present the main ideas of the proof and discuss the role played by the Davis building. This is joint work with Camille Horbez.

Ralf Köhl Topological twin buildings and topological Kac-Moody groups
Abstract: 2002 Linus published a paper in which he topologizes affine twin buildings. His axioms are so well-chosen that they allow also for a meaningful theory in the indefinite case, as Hartnick, Mars and R.K. have observed. Moreover, it is possible to topologize Kac-Moody groups via colimit techniques starting out with the Lie group topology on the spherical subgroups, as observed by Glöckner, Hartnick, Mars and R.K; here the Burns-Spatzier theorem enters in a subtle way. Once this is in place, one can study fundamental groups of Kac-Moody groups (as done by Harring and R.K.) and symmetric spaces for Kac-Moody groups (as done by Freyn, Horn, Hartnick, Vock and R.K.). In my talk I will outline how Linus' pioneering work on topological (twin) buildings made all this progress possible.

Norbert Knarr Embeddings of the doily and configuration theorems
Abstract: The doily is the smallest generalized quadrangle. We study embeddings of the doily into projective spaces and relate them to a newly discovered configuration theorem. It was shown by Jef Thas and Hendrik Van Maldeghem that in each 4-dimensional projective space there exists just one embedded doily, and each embedding of the doily into 3-dimensional spaces is obtained by projection. Using our configuration theorem, we construct embeddings of the doily into projective planes which are not obtained by projection from higher dimensional projective spaces. This is joint work with Markus Stroppel.

Alexander Lytchak Differential geometry and spherical buildings
Abstract: In the talk I would like to discuss applications of spherical buildings to Rank Rigidity and to Isoparametric Submanifolds.

Bernhard Mühlherr Generalized root systems
Abstract: One year ago, Dimitrov and Fioresi introduced generalized root systems. In their preprint they conjectured that each generalized root system is a quotient of a finite root system. It turns out that generalized root systems are closely related to Weyl groupoids. There is a classification of finite Weyl groupoids due to Cuntz and Heckenberger. In my talk I will outline a proof of the conjecture of Dimitrov and Fioresi that is based on the classification of finite Weyl groupoids. This is joint work with Michael Cuntz.

Damian Osajda Locally elliptic actions on nonpositively curved complexes
Abstract: This is joint work with Sergey Norin and Piotr Przytycki, and, in different part, with Thomas Haettel. We conjecture that locally elliptic actions of groups (that is, orbits of elements are bounded) on finite dimensional nonpositively curved complexes are elliptic (that is, orbits of the whole group are bounded). We prove this results for a variety of complexes including some Euclidean buildings. Such results have applications e.g. to automatic continuity. It is Linus Kramer, who explained to me the geometric motivation behind the latter notion.

Petra Schwer Chimney retractions
Abstract: One of the main tools when studying (affine) buildings are retractions. They are closely linked to both the combinatorics of folded galleries as well as the orbits of various prominent subgroups in the reductive groups defining the buildings. This link gives rise to a combinatorial model which allows to understand orbits on and sub-varieties of affine Grassmannians and affine flag varieties. In this talk I will not only introduce chimney retractions in affine buildings and explain the aforementioned connections – but I will also tell you why it is ultimatly Linus' fault that this research happened.

Jeroen Schillewaert A Tits alternative for Ã_{2} buildings
Abstract: In joint work with Jean Lécureux and Corentin Le Bars we prove a Tits alternative for Ã_{2 }buildings playing ping-pong with strongly regular hyperbolic elements. The latter are found using techniques to study random walks on CAT(0) spaces developed by Le Bars in his thesis.

Markus Stroppel Subalgebras of the octonion algebra
Abstract: We discuss the lattice of orbits of subalgebras of the split octonion algebra over an arbitrary field. The split Cayley hexagon is described in terms of subalgebras of dimensions 5 and 6, respectively. Applications to substructures in that hexagon are given. (The results are joint work with Norbert Knarr.)

Maneesh Thakur Embedding rank 2 normic tori in groups of type F_{4}
Abstract: In this talk, we will report on some results on embedding rank 2 normic tori in automorphism groups of Albert algebras. We shall also discuss a new characterisation of Albert algebras that are Tits first constructions. This is work in progress, with Holger Petersson.

Anne Thomas The geometry of conjugation in Euclidean isometry groups
Abstract: We give a simple and beautiful description of the geometry of conjugation within any split subgroup H of the full isometry group G of Euclidean space. We prove that for any h in H, the conjugacy class [h] is described geometrically by the move-set of its linearisation, while the set of elements conjugating h to a given h' in [h] is described by the fix-set of its linearisation. Examples include affine Coxeter groups, where we give finer results, certain crystallographic groups, and the group G itself. This is joint work with Elizabeth Milićević and Petra Schwer.

Hendrik Van Maldeghem Lines
Abstract: A line is a basic concept in most of axiomatic geometry. Geometric hyperplanes, on the other hand, are defined using the concept of a line. In this talk we reverse the roles of hyperplanes and lines and review some old and new results that characterise lines in Lie incidence geometries. We present some applications to spherical buildings.

Olga Varghese The isomorphism problem for Coxeter groups meets profinite rigidity
Abstract: By definition, a group is profinitely rigid if it is determined up to isomorphism by its finite quotients amongst all finitely generated residually finite groups. In this talk I will present recent work with Samuel Corson, Sam Hughes and Philip Möller on the question of profinite rigidity of Coxeter groups. I will also explain how profinite rigidity of Coxeter groups is connected to the isomorphism problem of these groups.

Richard Weiss Tits hexagons
Abstract: The notion of a Tits polygon generalizes the notion of a Moufang polygon. There is a natural construction that produces Tits polygons from a given spherical building and a matching Tits index. These are the Tits polygons of index type. We will describe ongoing efforts to understand the structure of the Tits hexagons of index type and their associated algebraic structures. This is joint work with Bernhard Mühlherr.

Stefan Witzel Exotic building lattices
Abstract: Most higher-rank euclidean buildings are Bruhat-Tits by the classification by Tits and Weiss, therefore lattices on them are arithmetic by work of Margulis. The exception is in dimension two where many non-arithmetic lattices exist, among them the simple lattices on products of trees found by Burger-Mozes. I will speak about efforts to find simple lattices on irreducible buildings. My share of effort is joint with Jean Lécureux and Thomas Titz Mite.

Poster session:
Paul Zellhofer – Kostant convexity for split real and complex Kac-Moody groups
Inga Valentiner-Branth – High dimensional Expanders from Kac-Moody Steinberg groups
Marjory Mwanza – On the construction of graph products from finite groups
Raquel Murat García – Fibrations over coset spaces and groups Cohomology
Bianca Marchionna – Invariants for tdlc groups acting cocompactly on buildings
Antonio Lopez Neumann – On conformal dimension of buildings
Amy Herron – Triangle presentations in Ã2 buildings
Sira Busch – Moufang property and groups of projectivities

Registration

Please register here to take part in the conference.
Deadline for registration: 23.08.2024

We plan to do a poster session and will send an email regarding contributions to all people, who have registered, after the registration deadline.

Support and child care

Childcare is available free of charge for all conference participants. We will ask for information during registration.

Venue and Travel Information

The conference takes place in lecture hall M3 (invited talks) on the ground floor of the lecture hall building, at

University of Münster Faculty of Mathematics and Computer science
Einsteinstrasse 62
48149 Münster
Germany

The postersession takes place in SRZ 216/217 on the second floor of the seminar building (Seminarraumzentrum, SRZ) next to the Faculty of Mathematics and Computer Science and the Cluster of Excellence Mathematics Münster, at