It is well known that basic properties of buildings can be deduced from their (Coxeter or, more particularly, Dynkin) diagrams. Such diagrams, furnished with encircled nodes, encircled orbits of symmetry groups, or even just encircled subsets of nodes, appear to be useful in many contexts. Bernhard is the real king of juggling with diagrams. In this talk I want to discuss some results on mainly spherical buildings that use some modest juggling with diagrams.
Generalised polygons play an important role in incidence geometry, building theory and graph theory. A (weak) generalised \(n\)-gon can be defined as a point-line geometry such that the incidence graph has diameter \(n\) and girth \(2n\). Here, we want to focus on generalised hexagons. Up to duality, only two classes of finite thick generalised hexagons are known: the split Cayley hexagons of order \((q,q)\) and the twisted triality hexagons of order \((q^3,q)\). Thas and Van Maldeghem characterised the natural embedding of the split Cayley hexagons in \(\text{PG}(6,q)\) using intersection numbers. Later, this was improved slightly by Ihringer.
In this talk we investigate the twisted triality hexagon, learn more about its natural embedding and obtain similar results. In particular we prove the following: A set of lines satisfies a list of properties (such as for example (Sd) every solid is incident with either 0, 1, \(q+1\) or \(2q+1\) lines of the set) if and only if it is the set of lines of a naturally embedded twisted triality hexagon.
Given a graph \(L\), a \((3,L)\)-complex is a simply connected polygonal complex in which every 2-cell is a triangle and the link of each vertex is isomorphic to \(L\).
We study \((3,L)\)-complexes under the assumption that the link \(L\) is a cubic bipartite graph which is at least 4–arc-transitive. We establish classification results beyond the flag-transitive case and prove that there exist exactly four face-transitive \((3,L)\)-complexes, among which precisely two are flag-transitive.
This confirms and extends observations of Caprace–Conder–Kaluba–Witzel on hyperbolic generalized triangle groups, and opens the way to constructing explicit examples of cocompact lattices in non-discrete totally disconnected locally compact groups.
When James Singer exhibited projective planes for all prime power orders in 1938, he realized these using the trace function of cubic extensions of a finite field and linked \(\text{trace}=0\) to perfect difference sets. In 1993, Cartwright, Mantero, Steger, and Zappa found that this trace function can be used to create a triangle presentation, which determines the structure of an \(\tilde{A}_2\) building. We recall the intrinsic connection between the perfect different sets of Singer and the triangle presentations of Cartwright et al. in order to translate the panel-regular groups of Essert (2013) and Witzel (2017) using triangle presentation nomenclature. This translation creates a uniform understanding of the panel-regular groups and vertex-regular groups via triangle presentations.
I present an elementary geometric proof for the fact that thick spherical buildings of types \(H_3\) and \(H_4\) cannot exist. This will be done by assuming their existence, giving direct constructions for root elations, which would imply the Moufang condition for generalized pentagons contained in them, and observing that this contradicts Jacques Tits’s Théorème 1 from his article “Non-existence de certains polygones généralisés.” The original non-existence proof was given by Tits in “Endliche Spiegelungsgruppen, die als Weylgruppen auftreten,” where he made use of his extension theorem, which is considered quite technical. Because of this, attempts have already been made to find a more elementary proof. This research formed part of a larger project on finding elementary geometric proofs of Tits’s results on thick irreducible spherical buildings of higher rank, which was suggested to me by Linus Kramer.
I will explain how to obtain symmetric presentations of \(\tilde{A}_2\) lattices by means of a simple combinatorial datum, which consists of a finite group \(G\), a subset \(S\) and a special permutation of \(S\). It turns out that this construction is quite flexible when \(G\) is abelian, thereby providing many interesting examples of exotic \(\tilde{A}_2\) lattices.
Ruelle-Taylor resonances play an important role in the study of dynamical systems. In this talk I will motivate the notion of Ruelle resonances and explain how the theory can be transferred to Weyl chamber flows on compact quotients of Euclidean buildings.
Let \(G\) be an affine or hyperbolic rank 2 Kac–Moody group over a finite field \(\mathbb{F}_q\), and let \(X\) denote the Tits building of \(G\). Suppose \(\Gamma\) is a non-uniform lattice in \(G\). When \(\Gamma\) is a standard parabolic subgroup for the negative BN-pair, we showed in earlier work, using the action of \(\Gamma\) on \(X\), that \(\Gamma\) is a lattice subgroup. In this talk, we will introduce Eisenstein series on \(\Gamma\backslash X\) for \(G\), discuss their convergence in a half-space via the Iwasawa decomposition of the Haar measure on \(G\), and establish their meromorphic continuation. This is joint work with Lisa Carbone and Paul Garrett.
An element in a group is said to be reciprocal if it is conjugate to its own inverse. In this talk, we will discuss about the classification of the reciprocal elements of the Hecke group. Subsequently, we discuss about the counting problem for reciprocal classes with respect to the word length in the context of Hecke groups.
A partial linear space is an incidence structure consisting of points and lines such that every line contains at least 3 points and every pair of points is in at most one line. I will assume partial linear spaces to be finite, not graphs nor linear spaces. Going all the way back to my PhD dissertation, I have been interested in partial linear spaces with varying degrees of symmetry, called \(k\)-ultrahomogeneity. The weakest level of symmetry among these is when \(k=2\): when the automorphism group is transitive on the ordered pairs of collinear points and on the ordered pairs of non-collinear points. This is equivalent to the automorphism group having rank 3.
Primitive rank 3 groups are classified. In 2005 I classified partial linear spaces admitting a primitive almost simple rank 3 group, and in 2008 the ones with a primitive rank 3 group of grid type. The case of primitive affine rank 3 groups was much harder to tackle, but was finally done in 2021, with Joanna Fawcett, John Bamberg and Cheryl Praeger (except for a few "hopeless" cases.)
Imprimitive rank 3 groups in general have not been classified, but in 2006, with Jonathan Hall, we managed to do the easiest case, when all lines have size 3. Recent work provided classification of imprimitive rank 3 groups with some extra assumptions. Rank 3 quasiprimitive groups were classified in 2011 (AD, Michael Giudici, Cai Heng Li, Geoffrey Pearce, Cheryl Praeger) and rank 3 innately transitive in 2023 (Anton Baykalov, AD and Cheryl Praeger). With Anton Baykalov and Cheryl Praeger, we have now classified the partial linear spaces with such groups, finding some nice infinite families and a small number of sporadic examples.
It is known by work of De Medts and Meulewaeter that simple Lie algebras over fields that are generated by extremal elements are G2-graded and parametrised by cubic norm pairs under natural conditions. It is also known by work of Wiedemann that F4-graded groups are parametrised by multiplicative conic alternative algebras over rings, a class of algebras that generalises composition algebras. We will show that for every cubic norm pair \(J\) over a ring, there exists a G2-graded Lie algebra \(L(J)\) that is parametrised by \(J\). Further, if \(J\) is a cubic Jordan matrix algebra over the multiplicative conic alternative algebra \(C\) (e.g., if \(J\) is a split Albert algebra over an octonion algebra \(C\)), then the G2-grading of \(L(J)\) can be refined to an F4-grading and the automorphism group of \(L(J)\) contains an F4-graded group that is parametrised by \(C\). This is joint work with Tom De Medts.
Right-angled Artin groups (RAAGs) form a family of finitely generated groups that plays a significant role in geometric group theory. Each of them acts geometrically on a CAT(0)-cube complex: the associated universal Salvetti complex.
We introduce a generalisation of RAAGs to topological groups based on the notion of generalised presentations. Those groups still admit a well-behaved action on a thicker version of the universal Salvetti complex, a cubed complex that displays some analogies with buildings. We use these constructions to create topological groups with specified finiteness properties.
Based on ongoing work with I. Castellano, B. Nucinkis, and Y. Santos Rego.
A BMW group is a group acting freely and transitively on the vertices of the Cartesian product of two trees. Newly discovered quaternionic arithmetic families of BMW groups will be described in the talk.
In this talk we discuss the structure and generation of some subgroups of Chevalley groups over non-Archemadian local fields. This is a joint work with Bertrand Rémy.
When \(K\) is a commutative division ring the point-hyperplane geometry of type \(A_{n,\{1,n\}}(K)\) admits a natural embedding in the vector space of traceless matrices of order \(n+1\). This embedding is dominant if and only if \(K\) is either algebraic over its prime subfield or perfect of positive characteristic; in this talk we shall provide an explicit description of its relatively universal cover for \(K\) an arbitrary field. We also observe that when \(K\) admits proper field automorphisms, the geometry \(A_{n,\{1,n\}}(K)\) does not admit the absolutely universal embedding. We conclude by briefly discussing the hyperplanes of \(A_{n,\{1,n\}}\) arising from the known dominant embeddings when \(K\) is a finite field.
A Kac-Moody group \(G\) generalises a split reductive group. By introducing a group ind-variety structure, one can define \(R\)-parabolic subgroups via a dynamic approach, and therefore a notion of \(G\)-complete reducibility. I will explore the behaviour of such \(R\)-parabolic subgroups and this theory. As a particular case, I will present a characterisation of \(G\)-completely reducible algebraic subgroups, recovering a result of D. Dawson for twin Euclidean buildings. We will also see that the dynamic approach to \(G\)-complete reducibility faithfully generalises a separate notion of 'complete reducibility' defined by P.-E. Caprace.
Let \(V\) be a finite-dimensional vector space over a field \(k\). The common basis complex \(CB(V)\) consists of sets of non-zero proper subspaces of \(V\) admitting a common basis. This complex was introduced by Rognes in 1992 as a “stable-building” model related to \(k\), whose homotopy orbits yield a filtration of the algebraic K-theory spectrum of \(k\). A key ingredient in this approach is the connectivity of \(CB(V)\), which would lead to simplified computations using such a filtration. Indeed, Rognes conjectured that the homology of \(CB(V)\) is concentrated in degree \(2\dim(V)-3\), a result established by Miller, Patzt, and Wilson in 2023. On the other hand, in joint work with Brück and Welker in 2024, we showed that \(CB(V)\) has the homotopy type of the poset \(PD(V)\) of partial direct sum decompositions of \(V\), whose dimension is also \(2\dim(V)-3\). The elements of this poset are sets of proper non-zero subspaces of \(V\) that are in internal direct sum, and the order is given by refinement. In fact, in an earlier work of Hanlon, Hersh, and Shareshian, it is proved that \(PD(V)\) is spherical when \(k\) is finite, thereby establishing Rognes’ conjecture using this homotopy equivalence.
In this talk, I will describe a generalisation of these complexes to the setting of spherical buildings. For a spherical building \(\Delta\), we define analogues \(CB(\Delta)\) and \(PD(\Delta)\), prove that they have the same (equivariant) homotopy type, and show that \(PD(\Delta)\) is spherical. Our result extends the vector-space case and yields an alternative proof of Rognes’ conjecture, independent of the one provided by Miller, Patzt and Wilson. This is based on joint work with John Shareshian and Volkmar Welker.
High-dimensional expanders (HDX) arise from generalizing the notion of expansion for graphs to (higher-dimensional) simplicial complexes. Like their one-dimensional versions, HDX have proven to be useful in theoretical computer science, but only a few constructions are known at this point. In this talk, I will give an introduction to the world of high-dimensional expansion. In particular, I will explain why spherical buildings and the complex opposite a fixed chamber in a spherical building are very good expanders. As usual for expanders, we are not just interested in single examples but infinite families of HDX of bounded degree. To this end, we start with an infinite complex constructed from a Kac-Moody-Steinberg (KMS) group, which is an amalgamated product of certain unipotent subgroups of Chevalley groups. The abundance of finite quotients of the KMS group gives rise to an infinite family of finite simplicial complexes that locally look like opposition complexes of spherical buildings. We will use the good expansion of the opposition complexes together with local-to-global results for high-dimensional expansion to show the desired outcome.
The talk will cover joint work with Izhar Oppenheim.
One effective way to investigate simple algebraic groups is through the study of related algebraic structures. Cubic norm structures, in particular, are useful when studying \(G_2\)-graded groups or Lie algebras. Similarly, structurable algebras, defined only over fields of characteristic different from 2 and 3, provide means to construct \(BC_1\)-graded algebras and groups. Hermitian cubic norm structures were originally introduced to study the class of skew-dimension one structurable algebras. Any such algebra can, after a quadratic extension, be constructed from a cubic norm structure. We will generalize the definition of a hermitian cubic norm structure, introduce the associated Lie algebra, and a subgroup of automorphisms of the Lie algebra. We show this group to be quite well behaved.
In this talk, we investigate a specific hyperbolic Kac–Moody group, and in particular focus on its subgroup \( U^+ \), generated by the positive root subgroups. We construct for \( U^+ \) finite simple quotients of Lie type of arbitrarily large rank, leading to the construction of high-dimensional expanders.
A simplicial arrangement is a set of linear hyperplanes decomposing the space into simplicial cones. More generally, a Tits arrangement decomposes a certain convex cone into simplicial cones. So far, Tits arrangements appeared (at least) in the following areas of mathematics:
In this talk we give an overview of the interplay between automorphisms and opposition in spherical buildings. In particular we will discuss the structure of the "opposite geometry" of an automorphism (those simplices mapped onto opposite simplices by the automorphism), and some general properties of the displacement spectrum. This is a summary of work over the past 10 years, all in joint work with Hendrik Van Maldeghem (amongst others).
Extended Weyl groups are reflection groups, and are divided in three different types depending of the underlying bilinear form. There are those of domestic, tubular and wild type. Those of domestic type are the simply laced affine Coxeter groups, while the tubular ones are elliptic Weyl groups. In the talk I will motivate why to study these groups, and will connect them to extended Coxeter groups, which were first been considered by Looijenga and van der Lek. And I will present some of their properties.
Harmonic analysis on symmetric spaces (such as Poincaré's hyperbolic half-plane) is a well-established subject, which has links with various fields: geometry, representation theory, number theory, etc. It is possible to develop a useful version of harmonic analysis on Euclidean buildings. Our main goal will be to introduce the most fundamental objects (Fourier transform), and then provide motivations and applications of the theory. If time permits, we will talk about partial differential equations on buildings and we will present applications for dynamical systems on some (true) manifolds. The talk will be based on joint work with J.-Ph. Anker and B. Trojan.
A permutation group is called 2-sharp if every 2-point-stabiliser is trivial. A finite 2-sharp transitive group is a Frobenius group, and hence has a unique regular subgroup, the Frobenius kernel. In the infinite case, however, things are quite different. An infinite 2-sharp transitive permuation groups does not have to possess a regular subgroup, and if there is one, it is not necessarily normal. Moreover, it may happen that there is more than one regular normal subgroup. In this talk I will discuss some results mainly concerning 2-sharp groups with regular abelian subgroups. As an application, I will show that there are wild nearfields whose multiplicative groups are metabelian.
We introduce the notion of a compactification-centric group: a topological group \(G\) is compactification-centric if in every semi-topological semi-group compactification of \(G\), one has \(sG=Gs\) for all \(s\).
For a local field \(k\) of characteristic zero, we show that the group of \(k\)-points of any algebraic group is compactification-centric. From this we deduce results concerning unitary representations of these groups.
Reid-Smith recently parametrised (P)-closed groups acting on trees using graph-based combinatorial structures known as local action diagrams. Properties of the acting (topological) group, such as being locally compact, compactly generated, discrete or simple, are reflected in its local action diagram. In this talk we outline how the scale values of a (P)-closed group can be determined from its local action diagram. This allows us to charactersise uniscalar and unimodular such groups.
Joint work with M. Chijoff and M. Ferov.
I will explain how ultrapowers of asymmetric metrics can be used to study compactifications and horofunctions. This is ongoing joint work with Corina Ciobotaru and Petra Schwer.
I will prove this result in detail using CAT(0) geometry and ultraproducts reducing it to the metrically complete case which was proved by Parreau in the early 2000s. Joint work with Jean Lécureux and Corentin Le Bars.
In this talk we report on recent progress on the classification of generating tuples of Fuchsian groups. While Louder showed that surface groups have unique Nielsen classes of irreducible generating tuples the situation is much more subtle and in general there are many such classes, all described in geometric terms by so called almost-orbifold covers endowed with rigid generating tuples. This is joint work with Ederson Dutra (Manaus).
Incidence geometries are in the basis of Tits buildings and related structures. Coset geometries are incidence structures derived from group cosets, where points, lines, and higher-dimensional elements correspond to cosets of certain subgroups. These capture symmetry and combinatorial properties of groups, particularly in relation to buildings and flag complexes.
In this talk, we will describe distinct ways of merging coset geometries by standard operations on groups, such as free product (with amalgamation), HNN-extension and split extension. Properties of coset geometries like flag-transitivity and residually connectedness are preserved under these operations. These constructions allow us to apply these operations to a graph of coset geometries, hinting to a Bass-Serre theory on coset geometries.
The author was partially supported by CMUP, member of LASI, which is financed by national funds through FCT – Fundação para a Ciência e a Tecnologia, I.P., under the projects with reference UIDB/00144/2025 and UIDP/00144/2025.
Wiener’s Tauberian theorem is a famous theorem in Fourier analysis which states that a function \(f \in L^1(\mathbb{R}^d)\) generates a dense ideal of \(L^1(\mathbb{R}^d)\) if and only if the Fourier transform of \(f\) vanishes nowhere. More generally, every locally compact group has a Fourier transform, and one can ask for which locally compact groups does the Fourier transform satisfy an analogue of Wiener’s theorem. Groups which satisfy an analogue of Wiener’s theorem are called Wiener groups. It was a big question in the mid-to-late 1900’s in Banach algebra theory to determine which locally compact groups are Wiener. For example, it is known that nilpotent groups and compactly generated groups of polynomial growth are Wiener. On the other hand, there are no known examples of non-amenable Wiener groups.
Essentially the only known class of group which are not Wiener are non-compact connected semisimple Lie groups. In this project we formulate a general result that can be used to determine when a non-amenable group with a Gelfand pair is not Wiener. We apply it to show that groups acting boundary transitively on a semi-regular tree are not Wiener, answering a question that has been open for some time. We also expect that groups acting strongly transitively on a Bruhat–Tits building are not Wiener, but at the time of writing this abstract, we have not had time to sort out the details of a proof of this. I will also discuss the relation of this project with the question of whether there exists a non-Wiener discrete group, which has been an open question for at least 50 years.
Abstract regular polytopes can be viewed as combinatorial structures that replicate the poset of faces of a concrete polytope. From the perspective of incidence geometry, they are thin, residually connected, flag-transitive geometries whose Buekenhout diagram is linear. Regular hypertopes arise from the same principles, but without requiring linearity of the diagram. Linear diagrams impose strong constraints on the geometry which in turn enables many interesting structural results to be proved. Surprisingly, it seems that sometimes the opposite is also true: allowing non-linear geometries into the picture can lead to new insights and greater clarity.
In the first part of the talk, I will present such a case by showing a full characterization of rank-three regular hypertopes whose automorphism group is a subgroup of \(PGL(2,q)\), where \(q\) is a prime power. In the second part, I will describe similarities between these hypertopes and tessellations of the real hyperbolic plane.
The finiteness properties \(\operatorname{F}_n\) and \(\operatorname{FP}_n\) generalize finite generation and finite presentability. In 2019, Skipper, Witzel and Zaremsky constructed a sequence \((G_n)_n\) of simple groups separated by finiteness properties, i.e. \(G_n\) is of type \(\operatorname{F}_{n-1}\) but not of type \(\operatorname{FP}_n\). Several equivalent generalizations of these finiteness properties have been developed for totally disconnected locally compact (tdlc) groups. We generalize the above result to the tdlc setting and construct a sequence \((G_n)_n\) of simple non-discrete tdlc groups separated by finiteness properties. Moreover, we construct a simple non-discrete tdlc group that is of type \(\operatorname{FP}_2\) but not compactly presented. Our examples arise from Smith groups, which act on biregular trees with prescribed local actions, together with Bestvina–Brady groups. This is joint work with Laura Bonn.
To any generalised Cartan matrix (GCM) \(A\) and any ring \(R\), Tits associated a Kac-Moody group \(\mathfrak{G}_A(R)\) defined by a presentation à la Steinberg. For a domain \(R\) with field of fractions \(\mathbb{K}\), we explore the question of whether the canonical map \(\varphi_R\colon\thinspace \mathfrak{G}_A(R)\to \mathfrak{G}_A(\mathbb{K})\) is injective. This question for Cartan matrices has a long history, and for GCMs was already present in Tits' foundational papers on Kac-Moody groups. We prove that for any 2-spherical GCM \(A\), the map \(\varphi_R\) is injective for all valuation rings \(R\) (under an additional minor condition (co)). This is joint work with Bernhard Mühlherr.
As a PhD student I was invited by Sergey Shpectorov to collaborate on a revision on Phan's theorems, to be used in the second generation proof of the classification of the finite simple groups. This became a team effort that eventually established all possible Phan-type theorems that can exist: An (Phan, Bennett-Shpectorov), Bn (Bennett, Hoffman, Horn, Nickel, Shpectorov, K.), Cn (Hoffman, Shpectorov, K.), Dn (Phan, Hoffman, Nickel, Shpectorov, K.), En (Phan, Devillers, Hoffman, Mühlherr, Shpectorov, Witzel, K.), F4 (Hoffman, Mühlherr, Sphectorov, Witzel, K.). It also linked to the study of centralizers of involutions in Kac-Moody-groups over finite fields for which Bernhard was the one to observe that these are lattices. Part of the Phan-type theorem team moved on to study their finite presentations (Devillers, Horn, Mühlherr, Witzel, K.), which in the affine case contributed to the study of finiteness properties of arithmetic lattices in semisimple Lie groups of positive characteristic (Bux, Witzel, K.), answering a question posed by Borel and Serre in 1976.
We will describe some recent results in the study of Tits-hexagons and their connection to Jordan algebras, Tits-diagrams and root systems. This is joint work with Bernhard.
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