Bruhat-Tits Buildings Research Articles

Positive crossratios, barycenters, trees and applications to maximal representations

We study metric properties of maximal framed representations of fundamental groups of surfaces in symplectic groups over real closed fields, interpreted as actions on Bruhat–Tits buildings endowed with adapted Finsler norms. We prove that the translation length can be computed as intersection with a geodesic current, give sufficient conditions guaranteeing that such a current is a multicurve, and, if the current is a measured lamination, construct an isometric embedding of the associated tree in the building. These results are obtained as application of more general results of independent interest on positive crossratios and actions with compatible barycenters.

Buildings, valuated matroids, and tropical linear spaces

AbstractAffine Bruhat–Tits buildings are geometric spaces extracting the combinatorics of algebraic groups. The building of parameterizes flags of subspaces/lattices in or, equivalently, norms on a fixed finite‐dimensional vector space, up to homothety. It has first been studied by Goldman and Iwahori as a piecewise‐linear analogue of symmetric spaces. The space of seminorms compactifies the space of norms and admits a natural surjective restriction map from the Berkovich analytification of projective space that factors the natural tropicalization map. Inspired by Payne's result that the analytification is the limit of all tropicalizations, we show that the space of seminorms is the limit of all tropicalized linear embeddings and prove a faithful tropicalization result for compactified linear spaces. The space of seminorms is in fact the tropical linear space associated to the universal realizable valuated matroid.

Bruhat-Tits buildings, representations of p-adic groups and Langlands correspondence

The Bruhat-Tits theory is a key ingredient in the construction of irreducible smooth representations of p-adic reductive groups. We describe generalizations to arbitrary such representations of several results recently obtained in the case of supercuspidal representations, in particular regarding the local Langlands correspondence and the internal structure of the L-packets.We prove that the enhanced L-parameters with semisimple cuspidal support are those which are obtained via the (ordinary) Springer correspondence.Let G be a connected reductive group over a non-archimedean field F of residual characteristic p. In the case where G splits over a tamely ramified extension of F and p does not divide the order of the Weyl group of G, we show that the enhanced L-parameters with semisimple cuspidal support correspond to irreducible smooth representations of G(F) with non-singular supercuspidal support via the local Langlands correspondence constructed by Kaletha, under the assumption that the latter satisfies certain expected properties of functorial nature. As a consequence, we obtain that every compound L-packet of G(F) contains at least one representation with non-singular supercuspidal support.

Wonderful compactifications of Bruhat–Tits buildings in the non-split case

Wonderful compactifications of Bruhat–Tits buildings in the non-split case

Isogeny graphs on superspecial abelian varieties: eigenvalues and connection to Bruhat–Tits buildings

Abstract We study for each fixed integer $g \ge 2$ , for all primes $\ell $ and p with $\ell \neq p$ , finite regular directed graphs associated with the set of equivalence classes of $\ell $ -marked principally polarized superspecial abelian varieties of dimension g in characteristic p, and show that the adjacency matrices have real eigenvalues with spectral gaps independent of p. This implies a rapid mixing property of natural random walks on the family of isogeny graphs beyond the elliptic curve case and suggests a potential construction of the Charles–Goren–Lauter-type cryptographic hash functions for abelian varieties. We give explicit lower bounds for the gaps in terms of the Kazhdan constant for the symplectic group when $g \ge 2$ . As a byproduct, we also show that the finite regular directed graphs constructed by Jordan and Zaytman also has the same property.

Applications of the Fixed Point Theorem for group actions on buildings to algebraic groups over polynomial rings

We apply the Fixed Point Theorem for the actions of finite groups on Bruhat–Tits buildings and their products to establish two results concerning the groups of points of reductive algebraic groups over polynomial rings in one variable, assuming that the base field is of characteristic zero. First, we prove that for a reductive k-group G, every finite subgroup of G(k[t]) is conjugate to a subgroup of G(k). This, in particular, implies that if k is a finite extension of the p-adic field Qp, then the group G(k[t]) has finitely many conjugacy classes of finite subgroups, which is a well-known property for arithmetic groups. Second, we give a short proof of the theorem of Raghunathan–Ramanathan [26] about G-torsors over the affine line.

Submodule codes as spherical codes in buildings

We give a generalization of subspace codes by means of codes of modules over finite commutative chain rings. We define a new class of Sperner codes and use results from extremal combinatorics to prove the optimality of such codes in different cases. Moreover, we explain the connection with Bruhat–Tits buildings and show how our codes are the buildings’ analogue of spherical codes in the Euclidean sense.

A new axiomatics for masures II

Abstract Masures are generalizations of Bruhat–Tits buildings. They were introduced by Gaussent and Rousseau in order to study Kac–Moody groups over valued fields. We prove that the intersection of two apartments of a masure is convex. Using this, we simplify the axiomatic definition of masures given by Rousseau.

An Intrinsic Characterization of Bruhat–Tits Buildings Inside Analytic Groups

Given a reductive group over a complete non-Archimedean field, it is well known that techniques from non-Archimedean analytic geometry provide an embedding of the corresponding Bruhat–Tits building into the analytic space associated with the group; by composing the embedding with maps to suitable analytic proper spaces, this eventually leads to various compactifications of the building. In the present paper, we give an intrinsic characterization of this embedding.

Injective metrics on buildings and symmetric spaces

In this article, we show that the Goldman–Iwahori metric on the space of all norms on a fixed vector space satisfies the Helly property for balls. On the non-Archimedean side, we deduce that most classical Bruhat–Tits buildings may be endowed with a natural piecewise ℓ ∞ $\ell ^\infty$ metric which is injective. We also prove that most classical semisimple groups over non-Archimedean local fields act properly and cocompactly on Helly graphs. This gives another proof of biautomaticity for their uniform lattices. On the Archimedean side, we deduce that most classical symmetric spaces of non-compact type may be endowed with a natural invariant Finsler metric, restricting to an ℓ ∞ $\ell ^\infty$ metric on each flat, which is coarsely injective. We also prove that most classical semisimple groups over Archimedean local fields act properly and cocompactly on injective metric spaces. We identify the injective hull of the symmetric space of GL ( n , R ) $\operatorname{GL}(n,\mathbb {R})$ as the space of all norms on R n $\mathbb {R}^n$ . The only exception is the special linear group: if n = 3 $n=3$ or n ⩾ 5 $n \geqslant 5$ and K $\mathbb {K}$ is a local field, we show that SL ( n , K ) $\operatorname{SL}(n,\mathbb {K})$ does not act properly and coboundedly on an injective metric space.

A Gallery Model for Affine Flag Varieties via Chimney Retractions

This paper provides a unified combinatorial framework to study orbits in certain affine flag varieties via the associated Bruhat–Tits buildings. We first formulate, for arbitrary affine buildings, the notion of a chimney retraction. This simultaneously generalizes the two well-known notions of retractions in affine buildings: retractions from chambers at infinity and retractions from alcoves. We then present a recursive formula for computing the images of certain minimal galleries in the building under chimney retractions, using purely combinatorial tools associated to the underlying affine Weyl group. Finally, for Bruhat–Tits buildings in the function field case, we relate these retractions and their effect on minimal galleries to double coset intersections in the corresponding affine flag variety.

Ramanujan Complexes and Golden Gates in PU(3)

In a seminal series of papers from the 80’s, Lubotzky, Phillips and Sarnak applied the Ramanujan–Petersson Conjecture for \(GL_{2}\) (Deligne’s theorem), to a special family of arithmetic lattices, which act simply-transitively on the Bruhat–Tits trees associated with \(SL_{2}({\mathbb {Q}}_{p})\). As a result, they obtained explicit Ramanujan Cayley graphs from \(PSL_{2}\left( {\mathbb {F}}_{p}\right) \), as well as optimal topological generators (“Golden Gates”) for the compact Lie group PU(2). In higher dimension, the naive generalization of the Ramanujan Conjecture fails, due to the phenomenon of endoscopic lifts. In this paper we overcome this problem for \(PU_{3}\) by constructing a family of arithmetic lattices which act simply-transitively on the Bruhat–Tits buildings associated with \(SL_{3}({\mathbb {Q}}_{p})\) and \(SU_{3}({\mathbb {Q}}_{p})\), while at the same time do not admit any representation which violates the Ramanujan Conjecture. This gives us Ramanujan complexes from \(PSL_{3}({\mathbb {F}}_{p})\) and \(PSU_{3}({\mathbb {F}}_{p})\), as well as golden gates for PU(3).

Typical representations via fixed point sets in Bruhat–Tits buildings

For a tame supercuspidal representation π \pi of a connected reductive p p -adic group G G , we establish two distinct and complementary sufficient conditions, formulated in terms of the geometry of the Bruhat–Tits building of G G , for the irreducible components of its restriction to a maximal compact subgroup to occur in a representation of G G which is not inertially equivalent to π \pi . The consequence is a set of broadly applicable tools for addressing the branching rules of π \pi and the unicity of [ G , π ] G [G,\pi ]_G -types.

Shadows in the Wild - Folded Galleries and Their Applications

This survey is about combinatorial objects related to reflection groups and their applications in representation theory and arithmetic geometry. Coxeter groups and folded galleries in Coxeter complexes are introduced in detail and illustrated by examples. Further it is explained how they relate to retractions in Bruhat-Tits buildings and to the geometry of affine flag varieties and affine Grassmannians. The goal is to make these topics accessible to a wide audience.

Degenerations of Grassmannians via Lattice Configurations

AbstractWe study degenerations of Grassmannians constructed using convex lattice configurations in Bruhat–Tits buildings. Using techniques from quiver representations, we analyze their special fibers, which are explicitly described as quiver Grassmannians. For a class of lattice configurations, called the locally linearly independent configurations, we show that our construction coincide with Mustafin degenerations, thus generalizing a result of Faltings. In such cases, our analysis of special fibers also generalizes results of Cartwright et al. As an application, we prove a smoothing criterion for limit linear series on arbitrary reducible nodal curves.

Free Flags Over Local Rings and Powering of High-dimensional Expanders

Abstract Powering the adjacency matrix of an expander graph results in a better expander of higher degree. In this paper we seek an analogue operation for high-dimensional (HD) expanders. We show that the naive approach to powering does not preserve HD expansion and define a new power operation, using geodesic walks on quotients of Bruhat–Tits buildings. Applying this operation results in HD expanders of higher degrees. The crux of the proof is a combinatorial study of flags of free modules over finite local rings. Their geometry describes links in the power complex, and showing that they are excellent expanders implies HD expansion for the power complex by Garland’s local-to-global technique. As an application, we use our power operation to obtain new efficient double samplers.

Filtrations and Buildings

We construct and study a scheme theoretical version of the Tits vectorial building, relate it to filtrations on fiber functors, and use them to clarify various constructions pertaining to Bruhat-Tits buildings, for which we also provide a Tannakian description.

STABILITY, COHOMOLOGY VANISHING, AND NONAPPROXIMABLE GROUPS

Several well-known open questions (such as: are all groups sofic/hyperlinear?) have a common form: can all groups be approximated by asymptotic homomorphisms into the symmetric groups$\text{Sym}(n)$(in the sofic case) or the finite-dimensional unitary groups$\text{U}(n)$(in the hyperlinear case)? In the case of$\text{U}(n)$, the question can be asked with respect to different metrics and norms. This paper answers, for the first time, one of these versions, showing that there exist finitely presented groups which arenotapproximated by$\text{U}(n)$with respect to the Frobenius norm$\Vert T\Vert _{\text{Frob}}=\sqrt{\sum _{i,j=1}^{n}|T_{ij}|^{2}},T=[T_{ij}]_{i,j=1}^{n}\in \text{M}_{n}(\mathbb{C})$. Our strategy is to show that some higher dimensional cohomology vanishing phenomena impliesstability, that is, every Frobenius-approximate homomorphism into finite-dimensional unitary groups is close to an actual homomorphism. This is combined with existence results of certain nonresidually finite central extensions of lattices in some simple$p$-adic Lie groups. These groups act on high-rank Bruhat–Tits buildings and satisfy the needed vanishing cohomology phenomenon and are thus stable and not Frobenius-approximated.

Common hyperbolic bases for chains of alternating or quadratic lattices

We give a short and purely bilinear proof of the fact that two chains of $p$-elementary lattices with quadratic form or alternating bilinear form over the $p$-adic integers ore more generally over a complete discrete valuation ring have common hyperbolic bases. This fact, which is useful for the study of Bruhat-Tits buildings, has been proven before with different methods by Abramenko and Nebe and by Frisch.

Completed Iwahori-Hecke algebras and parahoric Hecke algebras for Kac-Moody groups over local fields

Let G be a split Kac-Moody group over a non-Archimedean local field. We define a completion of the Iwahori-Hecke algebra of G, then we compute its center and prove that it is isomorphic (via the Satake isomorphism) to the spherical Hecke algebra of G. This is thus similar to the situation for reductive groups. Our main tool is the masure ℐ associated to this setting, which plays here the same role as Bruhat-Tits buildings do for reductive groups. In a second part, we associate a Hecke algebra to each spherical face F of type 0, extending a construction that was only known, in the Kac-Moody setting, for the spherical subgroup and for the Iwahori subgroup.