## Programme

Below is the planned programme for the upcoming Buildings conference, this schedule is preliminary and is subject to minor changes. Click here to download a copy of the Programme. Slides and notes from talks can be accessed here.

Wed, 06.10 |
Thu, 07.10 |
Fri, 08.10 |
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9:00 | 9:00 | Sebastian Bischof | |||

9:40 | Break | ||||

10:00 | Mima Stanojkovski | 10:20 | Amandine Escalier | ||

10:30 | Virtual Coffee Break (v) | 10:55 | Jeroen Schillewaert (v) | ||

11:20 | Maneesh Thakur (v) | 11:30 | Anne Thomas (v) | ||

11:55 | Philip Moeller | 12:10 | Lunch Break | ||

12:30 | Arrival and Registration | 12:30 | Lunch Break | ||

13:30 | Matthias Grüninger (v) | ||||

14:00 | Hendrik Van Maldeghem | 14:00 | Waltraud Lederle | 14:10 | Bernhard Mühlherr |

15:00 | Paulien Jansen | 15:00 | Annette Karrer | 15:00 | Goodbye Coffee |

15:40 | Coffee Break | 15:40 | Coffee Break | ||

16:15 | Eduard Schesler | 16:15 | Corina Ciobotaru (v) | 16:00 | The End |

16:50 | Michal Ferov | 16:50 | Lancelot Semal | ||

17:30 | Virtual Coffee Break (v) | 17:30 | End of Day 2 | ||

18:00 | End of Day 1 | ||||

DIY Dinner |

30 Minute Talk | 50 Minute Talk | (v) | Virtual event |

### In Person Events

**All in person events will run under the "3G" rule**. We expect all participants to be **fully recovered, fully vaccinated or negatively tested**. Please bring a test that is **not older than 24h at the start of the conference** if you don't meet at least one of the other criteria. We will check the 3G-requirement on Wednesday upon registration.

You can find a list of test centers in Magdeburg here, in case you prefer to take a test when you arrive. To stay safe during and after the conference **we ask you to check-in with the official "Corona-WarnApp"** whenever entering the lecture hall for the talks. The QR-codes to scan with your App will be located next to the entrance of the lecture hall where the talks take place. **The use of masks (medical, FFP2 or equivalent) is mandatory on our campus and inside all university buildings**, unless you are unable to wear a face mask due to health issues. Please make sure to bring enough masks for your stay here in Magdeburg.

### Virtual Events

All online talks will be given via Zoom, the links will be shared with all participants shortly before the start of the conference. We have also planned two virtual coffee breaks to give online and in person participants a chance to interact with each other: For these events we will use the platform wonder.me, an online space allowing visitors to speak via video chat in small groups. The link for this will also be shared prior to the conference and participants are encouraged to try this out in advance - this platform works best when using Google Chrome. **Don't forget to bring your own headset for these sessions!**

### DIY Dinner

Whilst there will be no official reception this year, we will provide a list of recommended restaurants and encourage in person participants to make their own dinner plans.

### Titles and Abstracts

Construction of RGD-systems over F_{2}

Let G be a group endowed with an RGD-system of type (W, S). Let U_{+} be the subgroup of G which is generated by all the root groups corresponding to positive roots. Then U_{+} is the direct limit of certain groups U_{w} with w ∊ W. Given suitable groups U_{w} for any w ∊ W, let U_{+} be the direct limit of these groups and assume that U_{w} → U_{+} is injective for any w ∊ W. In this talk we will show that under certain conditions the group U_{+} extends to an RGD-system. In such an RGD-system the root groups will have order 2.

Strong transitivity, Moufang’s condition and the Howe–Moore property

Firstly, we prove that every closed subgroup H of type-preserving automorphisms of a locally finite thick affine building ∆ of dimension ≥ 2 that acts strongly transitive on ∆ is Moufang. If moreover ∆ is irreducible and H is topologically simple, then we show H is the isotropic simple algebraic group over a non-Archimedean local field associated with ∆. Secondly, we generalizes the proof given in [Burger, Mozes, *Lattices in products of trees*, IHES Sci Publ Math, 2000] for the case of bi-regular trees to any locally finite thick affine building ∆, and proves that any topologically simple, closed, strongly transitive and type-preserving subgroup of Aut(∆) has the Howe–Moore property.

Local-to-Global rigidity of quasi-buildings

We say that a graph G is Local-to-Global rigid if there exists R>0 such that every other graph whose balls of radius R are isometric to the balls of radius R in G is covered by G. Examples include the Euclidean building of PSL_{n}(Q_{p}). We show that the rigidity of the building goes further by proving that a reconstruction is possible from only partial local information, called “print”. We use this to prove the rigidity of graphs quasi-isometric to the building among which are the torsion-free lattices of PSL_{n}(Q_{p}).

Automorphism groups of Cayley graphs of Coxeter groups - When are they discrete?

An automorphism group of a locally finite connected graph can be equipped with the permutation topology, which makes it a totally disconnected locally compact group. When the graph is a Cayley graph of a Coxeter group (with respect to the standard generating set), we give a characterisation, in terms of symmetries of the defining Coxeter system, when is the automorphism group non-discrete. Joint work with Federico Berlai

Moufang twin trees and ℤ-systems

A ℤ-system consists of a group X generated by a family of subgroups (X_{n})_{n∊ℤ} such that [X_{n},X_{m}] ≤ X_{n+1} ... X_{m-1} for all n < m and such that X admits an automorphism σ mapping X_{n} to X_{n+2} for all n. ℤ-systems naturally appear in Moufang twin trees. A theorem of Max Horn, Bernhard Mühlherr and myself states that if ( X, (X_{n})_{n∊ℤ}) is a ℤ-system such that X_{n} has order p ∊ P for all n, then X is nilpotent of class at most 2. This result can be generalised as follows: I will show in this talk that is also valid if X admits an automorphism T acting irreducibly on each subgroup X_{n}. This is joint work with Maximilian Parr.

Abelian Tits sets: a classification

Moufang sets were introduced by Jaqcues Tits over 30 years ago. It is however still an open conjecture that (proper) abelian Moufang sets are classified by Jordan division algebras. In this talk, we will discuss a generalization of Moufang sets, namely Tits sets. Under some mild conditions, we can prove that abelian Tits sets that are not Moufang correspond to simple Jordan algebras that are not division. This talk is based on work joined with Bernhard Mühlherr.

The boundary rigidity of lattices in products of trees

(joint work with Kasia Jankiewicz, Kim Ruane and Bakul Sathaye)

Each complete CAT(0) space has an associated topological space, called visual boundary that coincides with the Gromov boundary in case that the space is hyperbolic. A CAT(0) group G is called boundary rigid if the visual boundaries of all CAT(0) spaces admitting a geometric action by G are homeomorphic. If G is hyperbolic, G is boundary rigid. If G is not hyperbolic, G is not always boundary rigid. The first such example was found by Croke--Kleiner.

In this talk we will see that every group acting freely and cocompactly on a product of two regular trees of finite valence is boundary rigid. That means that every CAT(0) space that admits a geometric action of any such group has the boundary homeomorphic to a join of two copies of the Cantor set.

The minimal degree of a Cayley--Abels graph for Aut(T)

We show that the automorphism group of a d-regular tree can not act vertex-transitively with compact, open vertex stabilizers on a connected graph of degree smaller than d (unless d = 1). In most cases, this is just a consequence of simplicity of the alternating group on d-1 letters. This gives the special case d=5, which is surprisingly tricky.

Automorphisms of spherical buildings with nontrival opposition diagram

We discuss the extend to which we can classify the automorphisms of a spherical building with a given nontrivial opposition diagram. We pay particular attention to the case of a split building, for which the answer seems to be slightly nicer than the general case.

Automatic continuity for groups from GGT

An important result by Dudley from the 1960s states that every algebraic homomorphism from a locally compact Hausdorff group to a free (abelian) group is continuous. A generalization by Morris and Nickolas shows that this remains true for arbitrary free products of groups unless the image of the homomorphism is small. We are able to show this pattern remains true for many groups studied in GGT if their torsion subgroups are small. This is joint work with Daniel Keppeler and Olga Varghese.

Veldkamp Polygons and Tits Polygons

Veldkamp polygons are connected bipartite graphs in which the set of edges containing a given vertex is endowed with an opposition relation and which satisfy certain axioms that are variations of the axioms for a generalized polygon. In fact, a generalized n-gon is a Veldkamp n-gon for which the opposition relation at each vertex is trivial.For Veldkamp polygons there is a natural notion of the Moufang property and a Veldkamp polygon satisfying this property is called a Tits polygon.Over the last years we studied Tits polygons and produced several results. In my talk I intend to give a survey in which I explain our motivation for looking at these structures and explain some of our main results.This is joint work with Richard Weiss, Tufts University.

Rowing on the magic square

In this talk we will consider a couple of connections between the geometries of first and second row of the Freudenthal-Tits magic square.

Random subcomplexes of finite buildings and application to right-angled Coxeter groups

A group G is said to virtually algebraically fiber if G has a finite index subgroup H that admits an epimorphism to the integers with finitely generated kernel. The question of whether a group virtually algebraically fibers was originally motivated by Stallings fibering theorem which says that virtual algebraic fibering of the fundamental group certain 3-manifolds M is enough to deduce the existence of a topological fibering of a finite cover of M over the circle. A further class of groups for which the question of virtual algebraic fibering turned out to be a very rich one is the class of right-angled Coxeter groups (RACGs). In this talk I will discuss the notion of higher virtual algebraic fiberings, where finite generation is replaced by higher finiteness properties, with a special focus on RACGs whose defining flag complex is a finite building. We prove for particular classes of finite buildings that their random induced subcomplexes have a number of strong properties, most prominently that they are highly connected.From this we are able to deduce that the commutator subgroup of a RACG, with defining flag complex a finite building of a certain type, admits an epimorphism to the integers whose kernel has strong topological finiteness properties. The key tool we use is a generalization of an approach due to Jankiewicz-Norin-Wise involving Bestvina-Brady discrete Morse theory applied to the Davis complex of a RACG, together with some probabilistic arguments. This is joint work with Matthew Zaremsky.

Fixed points for group actions on 2-dimensional affine buildings

We prove a local-to-global result for fixed points of groups acting on 2-dimensional affine buildings. Joint work with Koen Struyve and Anne Thomas

The Type I conjecture on trees.

One of the most important family of groups appearing naturally in representation theory of locally compact groups is the family of Type I groups. Loosely speaking those groups are exactly the groups for which the irreducible representations are the "appropriate" building blocks of representations. Due to the importance of the concept it has been a very active domain of research to know whether a group is Type I or not. The question has been completely answered for discrete groups, connected Nilpotent Lie groups, reductive algebraic groups over non-Archimedean fields, adelic reductive groups, semisimple connected Lie groups. However, much less progress was achieved on non-discrete, non-linear groups. When it comes to those groups, the most advanced results so far concerns autmorphism groups of trees and it was conjectured by Nebbia in 1999 that a closed non-compact autmorphism group of trees is Type I if and only if it acts transitively on the boundary of the tree. This talk will be about the concepet of Type I groups and recent progress achievied in that conjecture.

Orders and polytropes: matrices from valuations

Let K be a discretely valued field with ring of integers R. To a d-by-d matrix M with integral coefficients one can associate an R-module, in K^{d}, and a polytope, in the Euclidean space of dimension d-1. We will look at the interplay between these two objects, from the point of view of tropical geometry and building on work of Plesken and Zassenhaus. This is joint work with Y. El Maazouz, M. A. Hahn, G. Nebe, and B. Sturmfels.

Cyclicity of Albert algebras

We will discuss an old problem posed by Adrian Albert, whether Albert (division) algebras contain a cubic cyclic subfield and its partial solution.

Divergence of geodesics in Coxeter groups

The divergence of a pair of geodesic rays is a measure of how fast they spread apart. For example, in Euclidean space the divergence of any pair of geodesic rays is linear, while in hyperbolic space it is exponential. In the 1980s Gersten used this idea to formulate a quasi-isometry invariant, also called divergence, which has been investigated for many important families of groups. We begin the study of divergence for arbitrary Coxeter groups, by formulating a combinatorial invariant of Coxeter systems called hypergraph index. We show that hypergraph index gives an upper bound on the divergence rate and conjecture that it gives a lower bound as well. This is joint work with Pallavi Dani, Yusra Naqvi, Dibyendu Roy and Ignat Soroko.