An indiscrete Bieberbach theorem: from amenable CAT ( 0 ) groups to Tits buildings

Tits buildings $${\Delta }_{\mathbb{Q}}(\mathbf{G})$$ of linear algebraic groupsGdefined over the field of rational numbers $$\mathbb{Q}$$ have played an important role in understanding partial compactifications of symmetric spaces and compactifications of locally symmetric spaces, cohomological properties of arithmetic subgroups and S-arithmetic subgroups of $$\mathbf{G}(\mathbb{Q})$$ . Curve complexes $$\mathcal{C}({S}_{g,n})$$ of surfaces S g,n were introduced to parametrize boundary components of partial compactifications of Teichmüller spaces and were later applied to understand properties of mapping class groups of surfaces and geometry and topology of 3-dimensional manifolds. Tits buildings are spherical buildings, and another important class of buildings consists of Euclidean buildings, for example, the Bruhat–Tits buildings of linear algebraic groups defined over local fields. In this paper, we summarize and compare some properties and applications of buildings and curve complexes. We try to emphasize their similarities but also point out differences. In some sense, curve complexes are combinations of spherical, Euclidean and hyperbolic buildings. We hope that such a comparison might motivate more questions and also suggest methods to solve them at the same time, and furthermore it might introduce buildings to people who study curve complexes and curve complexes to people who study buildings.

Let G be a connected semisimple group over a non-Archimedean local field. For every faithful, geometrically irreducible linear representation of G we define a compactification of the associated Bruhat–Tits building X(G). This yields a finite family of compactifications of X(G) which contains the polyhedral compactification. In addition, this family can be seen as a non-Archimedean analogue of the family of Satake compactifications for symmetric spaces.

Consider an affine Bruhat-Tits building Lat n of type An−1 and the complex distance in Lat n, i.e., the complete system of invariants of a pair of vertices of the building. An element of the Nazarov semigroup is a lattice in the duplicated p-adic space ℚ p n ⊕ ℚ p n . We investigate the behavior of the complex distance with respect to the natural action of the Nazarov semigroup on the building. Bibliography: 18 titles.

Let $G$ be a $p$-adic group that splits over an unramified extension. We decompose $\text{Rep}_{\unicode[STIX]{x1D6EC}}^{0}(G)$, the abelian category of smooth level $0$ representations of $G$ with coefficients in $\unicode[STIX]{x1D6EC}=\overline{\mathbb{Q}}_{\ell }$ or $\overline{\mathbb{Z}}_{\ell }$, into a product of subcategories indexed by inertial Langlands parameters. We construct these categories via systems of idempotents on the Bruhat–Tits building and Deligne–Lusztig theory. Then, we prove compatibilities with parabolic induction and restriction functors and the local Langlands correspondence.

Suppose $E$ is fully symmetric Banach function space on $(0,1)$ or $(0,\infty )$ or a fully symmetric Banach sequence space. We give necessary and sufficient conditions on $f\in E$ so that its orbit $\Omega (f)$ is the closed convex hull of its extreme po

Let G be a semisimple real Lie group of non-compact type, K a maximal compact subgroup and S G a semigroup with nonempty interior.We consider the ideal boundary (G/K) of the associated symmetric space and the flag manifolds G/P .We prove that the asymptotic imagewhere C is the maximal invariant control set for the action of S in the flag manifold G/P , with P a parabolic subgroup.The points that project over C are exactly the points of type in (Sx 0 ) (in the sense of the type of a cell in a Tits Building).

Motivated by applications in computational anatomy, we consider a second-order problem in the calculus of variations on object manifolds that are acted upon by Lie groups of smooth invertible transformations. This problem leads to solution curves known as Riemannian cubics on object manifolds that are endowed with normal metrics. The prime examples of such object manifolds are the symmetric spaces. We characterize the class of cubics on object manifolds that can be lifted horizontally to cubics on the group of transformations. Conversely, we show that certain types of non-horizontal geodesics on the group of transformations project to cubics. Finally, we apply second-order Lagrange--Poincar\'e reduction to the problem of Riemannian cubics on the group of transformations. This leads to a reduced form of the equations that reveals the obstruction for the projection of a cubic on a transformation group to again be a cubic on its object manifold.

Degeneration of point configurations in the projective plane

Let X be a higher-rank symmetric space or a Bruhat–Tits building of dimension at least 2 such that the isometry group of X has property (T). We prove that for every torsion-free lattice Γ⊂IsomX, any homology class in H1(Γ∖X,F2) admits a representative cycle of total length oX(Vol(Γ∖X)). As an application, we show that dimF2H1(Γ∖X,F2)=oX(Vol(Γ∖X)).

We discuss manifolds with positive sectimal curvature on which a group acts isometrically with one dimensional quotient. A number of the known examples have this property, but some potential families for new examples in dimension 7 arise as well. We discuss the geometry of these known examples and the connection that the candidates have with self-dual Einstein metrics. There are very few known examples of manifolds with positive sectional curvature. Apart from the compact rank one symmetric spaces, they exist only in dimensions 24 and below and are all obtained as quotients of a compact Lie group equipped with a biinvariant metric under an isometric group action. They consist of certain homogeneous spaces in dimensions 6,7,12,13, and 24 due to Berger [Be], Wallach [Wa], and Aloff–Wallach [AW], and of biquotients in dimensions 6,7, and 13 due to Eschenburg [E1],[E2],[E3] and Bazaikin [Ba]. When trying to find new examples, it is natural to search among manifolds with large isometry group, a program initiated by K. Grove in the 1990s, see [Wi] for a recent survey. Homogeneous spaces with positive curvature were classified in [Be],[Wa],[BB] in the 1970s. The next natural case to study is therefore manifolds on which a group acts isometrically with one dimensional quotient, so called co-homogeneity one manifolds. L. Verdiani [V1, V2] showed that in even dimensions, positively curved cohomogeneity one manifolds are equivariantly diffeomorphic to an isometric action on a rank one symmetric space. In odd dimensions, K. Grove and the author observed in 1998 that there are infinite families among the known nonsymmetric positively curved manifolds that admit isometric cohomogeneity one actions, and suggested a family of potential 7 dimensional candidates P k . In [GWZ], a classification in odd dimensions was carried out, and another family Q k and an isolated manifold R emerged in dimension 7. It is not yet known whether these manifolds admit a cohomogeneity one metric with positive curvature, although they all admit one with nonnegative curvature as a consequence of the main result in [GZ]. In [GWZ], the authors also discovered an intriguing connection that the manifolds P k and Q k have with a family of self-dual Einstein orbifold metrics constructed by Hitchin [Hi1] on S4. They naturally give rise to 3-Sasakian metrics on P k and Q k , which by definition have lots of positive curvature already. The purpose of this survey is threefold. In Section 2, we study the positively curved cohomogeneity one metrics on known examples with positive curvature, including the explicit functions that define the metric. In Section 3, we describe the classification theorem in [GWZ]. It is remarkable that among 7-manifolds where G = S3 ×S3 acts by cohomogeneity one, one has the known positively curved Eschenburg spaces E p , the Berger space B 7, the Aloff–Wallach space W 7, and the sphere S7, and that the candidates P k , Q k , and R all carry such an action as well. We thus carry out the proof in this most intriguing case where G = S3 ×S3 acts by cohomogeneity one on a compact 7-dimensional simply connected manifold. In Section 4, we describe the relationship to Hitchin’s self-dual Einstein metrics. We also discuss some curvature properties of these Einstein metrics and the metrics they define on P k and Q k . The behavior of these metrics, as well as the known metrics with positive curvature, are illustrated in a series of pictures.

In this paper, we deal with non-degenerate translators of the mean curvature flow in the well-known Einstein's static universe. We focus on the rotationally invariant translators, that is, those invariant by a natural isometric action of the special orthogonal group on the ambient space. In the classification list, there are three space-like cases and five time-like cases. All of them, except a totally geodesic example, have one or two conic singularities. Also, we show a uniqueness result based on the behaviour of the translator on its boundary. As an application, we extend an isometry of the sphere to the whole translator under simple conditions. This leads to a characterization of a bowl-like example.

Group actions and deformations for harmonic maps into symmetric spaces

A pseudo-Riemannian manifold $(M,g)$ is called a geodesic orbit manifold if any geodesic $\gamma$ of $M$ is an orbit of a $1$-parameter subgroup of the full isometry group of $(M,g)$. This terminology in the case of Riemannian manifolds was introduced in 1991 by O. Kowalski and L. Vanhecke, who initiated a systematic study of spaces $(M=G/H,g)$, where $G$ is an isometry group and $H$ is an isotropy subgroup. It should be noted that symmetric spaces, weakly symmetric spaces, naturally reductive homogeneous spaces, normal homogeneous spaces, generalized normal homogeneous spaces (but not only) are subclasses of geodesic orbit pseudo-Riemannian spaces. In this paper, we present examples of geodesic orbit pseudo-Riemannian manifolds. The examples are special $15$-dimensional pseudo $H$-type Lie groups, i.\,e., $2$-step nilpotent Lie groups of Heisenberg type equipped with a left invariant pseudo-Riemannian metric. To~construct the~corresponding examples, results on the~structure of the~Lie groups under consideration were used.

Polar actions with a fixed point

Let k be a nonarchimedean locally compact field of residue characteristic p, let G be a connected reductive group defined over k, let σ be an involutive k-automorphism of G, and H an open k-subgroup of the fixed points group of σ. We denote by G k and H k the groups of k-points of G and H. We obtain an analogue of the Cartan decomposition for the reductive symmetric space H k \Gk in the case where G is k-split and p is odd. More precisely, we obtain a decomposition of G k as a union of (H k , K)-double cosets, where K is the stabilizer of a special point in the Bruhat-Tits building of G over k. This decomposition is related to the H k -conjugacy classes of maximal σ-antiinvariant k-split tori in G. In a more general context, Benoist and Oh obtained a polar decomposition for any p-adic reductive symmetric space. In the case where G is k-split and p is odd, our decomposition makes more precise that of Benoist and Oh, and generalizes results of Offen for GL n .