Related papers: Compactifications of Bruhat-Tits buildings associa…

A self-affine tiling of a compact set G of positive Lebesgue measure is its partition to parallel shifts of a compact set which is affinely similar to G. We find all polyhedral sets (unions of finitely many convex polyhedra) that admit…

We define a compactification of symmetric spaces of noncompact type, seen as spaces of isometry classes of marked lattices, analogous to the Thurston compactification of the Teichm\"uller space, and we show that it is equivariantly…

We consider smooth representations of the unit group $G = \mathcal{A}^{\times}$ of a finite-dimensional split basic algebra $\mathcal{A}$ over a non-Archimedean local field. In particular, we prove a version of Gutkin's conjecture, namely,…

Equivariant compactifications of reductive groups can be described by combinatorial data. On the other hand, equivariant compactifications of the additive group G^n_a are more complicated in at least two respects. First, they often admit…

For all algebraic groups over non-Archimedean local fields, the bounded cohomology vanishes. This follows from the corresponding statement for automorphism groups of Bruhat--Tits buildings, which hinges on the solution to the flatmate…

Let K be a non-archimedean local field. This paper gives an explicit isomorphism between the dual of the special representation of GL_{n+1}(K)$and the space of harmonic cochains defined on the Bruhat-Tits building of GL_{n+1}(K), the…

Let $G$ be a connected reductive group over a $p$-adic local field $F$. R\'emy-Thuillier-Werner constructed embeddings of the (reduced) Bruhat-Tits building $\mathcal{B}(G,F)$ into the Berkovich spaces associated to suitable flag varieties…

Let $G$ be an almost simple, simply connected algebraic group defined over a number field $k$, and let $S$ be a finite set of places of $k$ including all infinite places. Let $X$ be the product over $v\in S$ of the symmetric spaces…

We study geodesically complete and locally compact Hadamard spaces X whose Tits boundary is a connected irreducible spherical building. We show that X is symmetric iff complete geodesics in X do not branch and a Euclidean building…

We show that for algebraic groups over local fields of characteristic zero, the following are equivalent: Every homomorphism has a closed image, every unitary representation decomposes into a direct sum of finite-dimensional and mixing…

We prove a purely topological characterization of the Moufang property for disconnected compact polygons in terms of convergence groups. As a consequence, we recover the fact that a locally finite thick affine building of rank 3 is a…

Let F be a non-Archimedean local field and let E be a finite extension of F. Let G be a split semisimple F group. We discuss how to compare volumes on the Bruhat-Tits buildings B_E and B_F of G(E) and G(F) respectively.

We give a geometric interpretation of the maximal Satake compactification of symmetric spaces $X=G/K$ of noncompact type, showing that it arises by attaching the horofunction boundary for a suitable $G$-invariant Finsler metric on $X$. As…

Let G be a real semisimple Lie group with finite center, with a finite number of connected components and without compact factor. We are interested in the homogeneous space of Cartan subgroups of G, which can be also seen as the space of…

To any graph $G$ one can associate a toric variety $X(\mathcal{P}G)$, obtained as a blowup of projective space along coordinate subspaces corresponding to connected subgraphs of $G$. The polytope of this toric variety is the graph…

We compute the low degree $\ell$-adic intersection cohomology of symplectic local systems on the Satake compactification of the moduli space $A_g$ of principally polarized abelian varieties. We prove that only a small finite list of…

We obtain an analog of the compression of angles theorem in symmetric spaces for Bruhat--Tits buildings of the type $A$. More precisely, consider a $p$-adic linear space $V$ and the set $Lat(V)$ of all lattices in $V$. The complex distance…

Let $G$ be a locally compact group. Then for every $G$-space $X$ the maximal $G$-proximity $\beta_G$ can be characterized by the maximal topological proximity $\beta$ as follows: $$ A \ \overline{\beta_G} \ B \Leftrightarrow \exists V \in…

We show that there exists a quasi-isometric embedding of the product of $n$ copies of $\mathbb{H}_{\mathbb{R}}^2$ into any symmetric space of non-compact type of rank $n$, and there exists a bi-Lipschitz embedding of the product of $n$…

Inspired by Manin's approach towards a geometric interpretation of Arakelov theory at infinity, we interpret in this paper non-Archimedean local intersection numbers of linear cycles in $P^{n-1}$ with the combinatorial geometry of the…