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Suppose G is an amenable locally compact group with lattice subgroup
$\Gamma $. Grosvenor [‘A relation between invariant means on Lie groups and invariant means on their discrete subgroups’, Trans. Amer. Math. Soc.288(2) (1985), 813–825] showed that there is a natural affine injection
$\iota : {\text {LIM}}(\Gamma )\to {\text {TLIM}}(G)$ and that
$\iota $ is a surjection essentially in the case
$G={\mathbb R}^d$,
$\Gamma ={\mathbb Z}^d$. In the present paper it is shown that
$\iota $ is a surjection if and only if
$G/\Gamma $ is compact.
Let
${\mathbf {G}}$
be a semisimple algebraic group over a number field K,
$\mathcal {S}$
a finite set of places of K,
$K_{\mathcal {S}}$
the direct product of the completions
$K_{v}, v \in \mathcal {S}$
, and
${\mathcal O}$
the ring of
$\mathcal {S}$
-integers of K. Let
$G = {\mathbf {G}}(K_{\mathcal {S}})$
,
$\Gamma = {\mathbf {G}}({\mathcal O})$
and
$\pi :G \rightarrow G/\Gamma $
the quotient map. We describe the closures of the locally divergent orbits
${T\pi (g)}$
where T is a maximal
$K_{\mathcal {S}}$
-split torus in G. If
$\# S = 2$
then the closure
$\overline {T\pi (g)}$
is a finite union of T-orbits stratified in terms of parabolic subgroups of
${\mathbf {G}} \times {\mathbf {G}}$
and, consequently,
$\overline {T\pi (g)}$
is homogeneous (i.e.
$\overline {T\pi (g)}= H\pi (g)$
for a subgroup H of G) if and only if
${T\pi (g)}$
is closed. On the other hand, if
$\# \mathcal {S}> 2$
and K is not a
$\mathrm {CM}$
-field then
$\overline {T\pi (g)}$
is homogeneous for
${\mathbf {G}} = \mathbf {SL}_{n}$
and, generally, non-homogeneous but squeezed between closed orbits of two reductive subgroups of equal semisimple K-ranks for
${\mathbf {G}} \neq \mathbf {SL}_{n}$
. As an application, we prove that
$\overline {f({\mathcal O}^{n})} = K_{\mathcal {S}}$
for the class of non-rational locally K-decomposable homogeneous forms
$f \in K_{\mathcal {S}}[x_1, \ldots , x_{n}]$
.
We consider autocorrelation functions for supersymmetric quantum mechanical systems (consisting of a fermion and a boson) confined in trigonometric Pöschl–Teller partner potentials. We study the limit of rescaled autocorrelation functions (at random time) as the localization of the initial state goes to infinity. The limiting distribution can be described using pairs of Jacobi theta functions on a suitably defined homogeneous space, as a corollary of the work of Cellarosi and Marklof. A construction by Contreras-Astorga and Fernández provides large classes of Pöschl-Teller partner potentials to which our analysis applies.
A subset X of a Polish group G is Haar null if there exists a Borel probability measure μ and a Borel set B containing X such that μ(gBh) = 0 for every g, h ∈ G. A set X is Haar meager if there exists a compact metric space K, a continuous function f : K → G and a Borel set B containing X such that f−1(gBh) is meager in K for every g, h ∈ G. We calculate (in ZFC) the four cardinal invariants (add, cov, non, cof) of these two σ-ideals for the simplest non-locally compact Polish group, namely in the case $G = \mathbb {Z}^\omega$. In fact, most results work for separable Banach spaces as well, and many results work for Polish groups admitting a two-sided invariant metric. This answers a question of the first named author and Vidnyánszky.
We prove several general conditional convergence results on ergodic averages for horocycle and geodesic subgroups of any continuous
$\operatorname {SL}(2, \mathbb {R})$
-action on a locally compact space. These results are motivated by theorems of Eskin, Mirzakhani and Mohammadi on the
$\operatorname {SL}(2, \mathbb {R})$
-action on the moduli space of Abelian differentials. By our argument we can derive from these theorems an improved version of the ‘weak convergence’ of push-forwards of horocycle measures under the geodesic flow and a short proof of weaker versions of theorems of Chaika and Eskin on Birkhoff genericity and Oseledets regularity in almost all directions for the Teichmüller geodesic flow.
A measure on a locally compact group is said to be spread out if one of its convolution powers is not singular with respect to Haar measure. Using Markov chain theory, we conduct a detailed analysis of random walks on homogeneous spaces with spread out increment distribution. For finite volume spaces, we arrive at a complete picture of the asymptotics of the n-step distributions: they equidistribute towards Haar measure, often exponentially fast and locally uniformly in the starting position. In addition, many classical limit theorems are shown to hold. In the infinite volume case, we prove recurrence and a ratio limit theorem for symmetric spread out random walks on homogeneous spaces of at most quadratic growth. This settles one direction in a long-standing conjecture.
Let
$\unicode[STIX]{x1D6F4}$
be a compact orientable surface of genus
$g=1$
with
$n=1$
boundary component. The mapping class group
$\unicode[STIX]{x1D6E4}$
of
$\unicode[STIX]{x1D6F4}$
acts on the
$\mathsf{SU}(3)$
-character variety of
$\unicode[STIX]{x1D6F4}$
. We show that the action is ergodic with respect to the natural symplectic measure on the character variety.
Furstenberg has associated to every topological group $G$ a universal boundary $\unicode[STIX]{x2202}(G)$. If we consider in addition a subgroup $H<G$, the relative notion of $(G,H)$-boundaries admits again a maximal object $\unicode[STIX]{x2202}(G,H)$. In the case of discrete groups, an equivalent notion was introduced by Bearden and Kalantar (Topological boundaries of unitary representations. Preprint, 2019, arXiv:1901.10937v1) as a very special instance of their constructions. However, the analogous universality does not always hold, even for discrete groups. On the other hand, it does hold in the affine reformulation in terms of convex compact sets, which admits a universal simplex $\unicode[STIX]{x1D6E5}(G,H)$, namely the simplex of measures on $\unicode[STIX]{x2202}(G,H)$. We determine the boundary $\unicode[STIX]{x2202}(G,H)$ in a number of cases, highlighting properties that might appear unexpected.
In this article, we study compactifications of homogeneous spaces coming from equivariant, open embeddings into a generalized flag manifold $G/P$. The key to this approach is that in each case $G/P$ is the homogeneous model for a parabolic geometry; the theory of such geometries provides a large supply of geometric tools and invariant differential operators that can be used for this study. A classical theorem of Wolf shows that any involutive automorphism of a semisimple Lie group $G$ with fixed point group $H$ gives rise to a large family of such compactifications of homogeneous spaces of $H$. Most examples of (classical) Riemannian symmetric spaces as well as many non-symmetric examples arise in this way. A specific feature of the approach is that any compactification of that type comes with the notion of ‘curved analog’ to which the tools we develop also apply. The model example of this is a general Poincaré–Einstein manifold forming the curved analog of the conformal compactification of hyperbolic space. In the first part of the article, we derive general tools for the analysis of such compactifications. In the second part, we analyze two families of examples in detail, which in particular contain compactifications of the symmetric spaces $\mathit{SL}(n,\mathbb{R})/\mathit{SO}(p,n-p)$ and $\mathit{SO}(n,\mathbb{C})/\mathit{SO}(n)$. We describe the decomposition of the compactification into orbits, show how orbit closures can be described as the zero sets of smooth solutions to certain invariant differential operators and prove a local slice theorem around each orbit in these examples.
We define a natural topology on the collection of (equivalence classes up to scaling of) locally finite measures on a homogeneous space and prove that in this topology, pushforwards of certain infinite-volume orbits equidistribute in the ambient space. As an application of our results we prove an asymptotic formula for the number of integral points in a ball on some varieties as the radius goes to infinity.
This paper concerns the study of the global structure of measure-preserving actions of countable groups on standard probability spaces. Weak containment is a hierarchical notion of complexity of such actions, motivated by an analogous concept in the theory of unitary representations. This concept gives rise to an associated notion of equivalence of actions, called weak equivalence, which is much coarser than the notion of isomorphism (conjugacy). It is well understood now that, in general, isomorphism is a very complex notion, a fact which manifests itself, for example, in the lack of any reasonable structure in the space of actions modulo isomorphism. On the other hand, the space of weak equivalence classes is quite well behaved. Another interesting fact that relates to the study of weak containment is that many important parameters associated with actions, such as the type, cost, and combinatorial parameters, turn out to be invariants of weak equivalence and in fact exhibit desirable monotonicity properties with respect to the pre-order of weak containment, a fact that can be useful in certain applications. There has been quite a lot of activity in this area in the last few years, and our goal in this paper is to provide a survey of this work.
We consider point distributions in compact connected two-point homogeneous spaces (Riemannian symmetric spaces of rank one). All such spaces are known: they are the spheres in the Euclidean spaces, the real, complex and quaternionic projective spaces and the octonionic projective plane. For all such spaces the best possible bounds for the quadratic discrepancies and sums of pairwise distances are obtained in the paper (Theorems 2.1 and 2.2). Distributions of points of
$t$
-designs on such spaces are also considered (Theorem 2.3). In particular, it is shown that the optimal
$t$
-designs meet the best possible bounds for quadratic discrepancies and sums of pairwise distances (Corollary 2.1). Our approach is based on the Fourier analysis on two-point homogeneous spaces and explicit spherical function expansions for discrepancies and sums of distances (Theorems 4.1 and 4.2).
Let
$X$
be a solenoid, i.e. a compact, finite-dimensional, connected abelian group with normalized Haar measure
$\unicode[STIX]{x1D707}$
, and let
$\unicode[STIX]{x1D6E4}\rightarrow \operatorname{Aff}(X)$
be an action of a countable discrete group
$\unicode[STIX]{x1D6E4}$
by continuous affine transformations of
$X$
. We show that the probability measure preserving action
$\unicode[STIX]{x1D6E4}\curvearrowright (X,\unicode[STIX]{x1D707})$
does not have the spectral gap property if and only if there exists a
$p_{\text{a}}(\unicode[STIX]{x1D6E4})$
-invariant proper subsolenoid
$Y$
of
$X$
such that the image of
$\unicode[STIX]{x1D6E4}$
in
$\operatorname{Aff}(X/Y)$
is a virtually solvable group, where
$p_{\text{a}}:\operatorname{Aff}(X)\rightarrow \operatorname{Aut}(X)$
is the canonical projection. When
$\unicode[STIX]{x1D6E4}$
is finitely generated or when
$X$
is the
$a$
-adic solenoid for an integer
$a\geq 1$
, the subsolenoid
$Y$
can be chosen so that the image
$\unicode[STIX]{x1D6E4}$
in
$\operatorname{Aff}(X/Y)$
is a virtually abelian group. In particular, an action
$\unicode[STIX]{x1D6E4}\curvearrowright (X,\unicode[STIX]{x1D707})$
by affine transformations on a solenoid
$X$
has the spectral gap property if and only if
$\unicode[STIX]{x1D6E4}\curvearrowright (X,\unicode[STIX]{x1D707})$
is strongly ergodic.
We generalize Skriganov’s notion of weak admissibility for lattices to include standard lattices occurring in Diophantine approximation and algebraic number theory, and we prove estimates for the number of lattice points in sets such as aligned boxes. Our result improves on Skriganov’s celebrated counting result if the box is sufficiently distorted, the lattice is not admissible, and, e.g., symplectic or orthogonal. We establish a criterion under which our error term is sharp, and we provide examples in dimensions
$2$
and
$3$
using continued fractions. We also establish a similar counting result for primitive lattice points, and apply the latter to the classical problem of Diophantine approximation with primitive points as studied by Chalk, Erdős, and others. Finally, we use o-minimality to describe large classes of sets to which our counting results apply.
Let $G$ be a real reductive group and $Z=G/H$ a unimodular homogeneous $G$ space. The space $Z$ is said to satisfy VAI (vanishing at infinity) if all smooth vectors in the Banach representations $L^{p}(Z)$ vanish at infinity, $1\leqslant p<\infty$. For $H$ connected we show that $Z$ satisfies VAI if and only if it is of reductive type.
The classical theorem of Vizing states that every graph of maximum degree $d$ admits an edge coloring with at most $d+1$ colors. Furthermore, as it was earlier shown by Kőnig, $d$ colors suffice if the graph is bipartite. We investigate the existence of measurable edge colorings for graphings (or measure-preserving graphs). A graphing is an analytic generalization of a bounded-degree graph that appears in various areas, such as sparse graph limits, orbit equivalence and measurable group theory. We show that every graphing of maximum degree $d$ admits a measurable edge coloring with $d+O(\sqrt{d})$ colors; furthermore, if the graphing has no odd cycles, then $d+1$ colors suffice. In fact, if a certain conjecture about finite graphs that strengthens Vizing’s theorem is true, then our method will show that $d+1$ colors are always enough.
Let $G$ be an algebraic real reductive group and $Z$ a real spherical $G$-variety, that is, it admits an open orbit for a minimal parabolic subgroup $P$. We prove a local structure theorem for $Z$. In the simplest case where $Z$ is homogeneous, the theorem provides an isomorphism of the open $P$-orbit with a bundle $Q\times _{L}S$. Here $Q$ is a parabolic subgroup with Levi decomposition $L\ltimes U$, and $S$ is a homogeneous space for a quotient $D=L/L_{n}$ of $L$, where $L_{n}\subseteq L$ is normal, such that $D$ is compact modulo center.
The goal of this paper is the study of homogeneous Riemannian structure tensors within the framework of reduction under a group H of isometries. In a first result, H is a normal subgroup of the group of symmetries associated with the reducing tensor . The situation when H is any group acting freely is analyzed in a second result. The invariant classes of homogeneous tensors are also investigated when reduction is performed. It turns out that the geometry of the fibres is involved in the preservation of some of them. Some classical examples illustrate the theory. Finally, the reduction procedure is applied to fibrings of almost contact manifolds over almost Hermitian manifolds. If the structure is, moreover, Sasakian, the obtained reduced tensor is homogeneous Kähler.
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