Corank and asymptotic filling-invariants for symmetric spaces

Let X be a Riemannian symmetric space of noncompact type. We prove that there exists an embedded submanifold which is quasi-isometric to a manifold with strictly negative sectional curvature, which intersects a given flat F in a geodesic line and which satisfies dim(Y) — 1 = dim(X) — rank(X). This yields an estimate of the hyperbolic corank of X. As another application we show that certain asymptotic filling invariants of X are exponential. Submitted: February 1999, Revised version: November 1999.     

Patterson-Sullivan Theory in Higher Rank Symmetric Spaces

Let X = G/K be a Riemannian symmetric space of noncompact type and a discrete “generic” subgroup of G with critical exponent . Denote by the set of regular elements of the geometric boundary of X. We show that the support of all -invariant conformal densities of dimension on (e.g. Patterson-Sullivan densities) lie in a same and single regular G-orbit on . This provides information on the large-scale growth of -orbits in X. If in addition we assume to be of divergence type, then there is a unique density of the previous type. Furthermore, we explicitly determine and this G-orbit for lattices, and show that they are of divergence type. Submitted: November 1997, revised: January 1999.     

Asymptotically invariant sequences and an action of SL (2,Z) on the 2-sphere

LetG be a countable group which acts non-singularly and ergodically on a Lebesgue space (X, ȑ, μ). A sequence (B _n) in ℒ is calledasymptotically invariant in lim _n μ (B _nΔgB _n)=0 for everygεG. In this paper we show that the existence of such sequences can be characterized by certain simple assumptions on the cohomology of the action ofG onX. As an explicit example we prove that a natural action of SL (2,Z) on the 2-sphere has no asymptotically invariant sequences. The last section deals with a particular cocycle for this action which has an interpretation as a random walk on the integers with “time” in SL (2,Z).     

Cantor set and interpolation

In 1998, Y. Benyamini published interesting results concerning interpolation of sequences using continuous functions ℝ → ℝ. In particular, he proved that there exists a continuous function ℝ → ℝ which in some sense “interpolates” all sequences (x _n ) _n∈ℤ ∈ [0, 1]^ℤ “simultaneously.” In 2005, M.R. Naulin and C. Uzcátegui unified and generalized Benyamini’s results. In this paper, the case of topological spaces X and Y with an Abelian group acting on X is considered. A similar problem of “simultaneous interpolation” of all “generalized sequences” using continuous mappings X → Y is posed. Further generalizations of Naulin-Uncátegui theorems, in particular, multidimensional analogues of Benyamini’s results are obtained.     

Distribution results for lattices in SL(2,ℚ p )

In this paper we study the ergodic properties of the linear action of lattices Γ of SL(2,ℚ_p) on ℚ_p × ℚ_p and distribution results for orbits of Γ. Following Serre, one can define a “geodesic flow” for an associated tree (actually associated to GL(2,ℚ_p)). The approach we use is based on an extension of this approach to “frame flows” which are a natural compact group extension of the geodesic flow.     

The sweet mystery of compatibilism

Any satisfactory account of freedom must capture, or at least permit, the mysteriousness of freedom—a “sweet” mystery involving a certain kind of ignorance rather than a “sour” mystery of unintelligibility, incoherence, or unjustifiedness. I argue that compatibilism can capture the sweet mystery of freedom. I argue first that an action is free if and only if a certain “rationality constraint” is satisfied, and that nothing in standard libertarian accounts of freedom entails its satisfaction. Satisfaction of this constraint is consistent with the universal causal predetermination of action (UCP). If UCP is true and the rationality constraint satisfied, there’s a sense in which our actions are explanatorily (though not necessarily causally) overdetermined. While it seems plausible (given UCP) that our actions are so overdetermined, it seems utterly mysterious why they should be so overdetermined. Compatibilism’s capacity to accommodate this mystery is a mark in its favor.     

Group measure space decomposition of II1 factors and W*-superrigidity

We prove a “unique crossed product decomposition” result for group measure space II₁ factors L ^∞(X)⋊Γ arising from arbitrary free ergodic probability measure preserving (p.m.p.) actions of groups Γ in a fairly large family G\mathcal{G}, which contains all free products of a Kazhdan group and a non-trivial group, as well as certain amalgamated free products over an amenable subgroup. We deduce that if T _n denotes the group of upper triangular matrices in PSL (n,ℤ), then any free, mixing p.m.p. action of G = \operatornamePSL(n,\mathbbZ)*_Tn\operatornamePSL(n,\mathbbZ)\Gamma=\operatorname{PSL}(n,\mathbb{Z})*_{T_{n}}\operatorname{PSL}(n,\mathbb{Z}) is W^∗-superrigid, i.e. any isomorphism between L ^∞(X)⋊Γ and an arbitrary group measure space factor L ^∞(Y)⋊Λ, comes from a conjugacy of the actions. We also prove that for many groups Γ in the family G\mathcal{G}, the Bernoulli actions of Γ are W^∗-superrigid.     

On log canonical divisors that are log quasi-numerically positive

Let (X Δ) be a four-dimensional log variety that is projective over the field of complex numbers. Assume that (X, Δ) is not Kawamata log terminal (klt) but divisorial log terminal (dlt). First we introduce the notion of “log quasi-numerically positive”, by relaxing that of “numerically positive”. Next we prove that, if the log canonical divisorK _X+Δ is log quasi-numerically positive on (X, Δ) then it is semi-ample.     

On a spectral sequence for equivariant K-theory

We apply the “homotopy coniveau” machinery developed by the first-named author to the K-theory of coherent G-sheaves on a finite type G-scheme X over a field, where G is a finite group. This leads to a definition of G-equivariant higher Chow groups (different from the Chow groups of classifying spaces constructed by Totaro and generalized to arbitrary X by Edidin–Graham) and an Atiyah–Hirzebruch spectral sequence from the G-equivariant higher Chow groups to the higher K-theory of coherent G-sheaves on X. This spectral sequence generalizes the spectral sequence from motivic cohomology to K-theory constructed by Bloch–Lichtenbaum and Friedlander–Suslin. The first-named author gratefully acknowledges the support of the Humboldt Foundation through the Wolfgang Paul Program, and support of the NSF via grants DMS-0140445 and DMS-0457195.     

Gromov hyperbolic cubic graphs

If X is a geodesic metric space and x ₁; x ₂; x ₃ ∈ X, a geodesic triangle T = {x ₁; x ₂; x ₃} is the union of the three geodesics [x ₁ x ₂], [x ₂ x ₃] and [x ₃ x ₁] in X. The space X is δ-hyperbolic (in the Gromov sense) if any side of T is contained in a δ-neighborhood of the union of the two other sides, for every geodesic triangle T in X. We denote by δ(X) the sharp hyperbolicity constant of X, i.e., δ(X) = inf {δ ≥ 0: X is δ-hyperbolic}. We obtain information about the hyperbolicity constant of cubic graphs (graphs with all of their vertices of degree 3), and prove that for any graph G with bounded degree there exists a cubic graph G* such that G is hyperbolic if and only if G* is hyperbolic. Moreover, we prove that for any cubic graph G with n vertices, we have δ(G) ≤ min {3n/16 + 1; n/4}. We characterize the cubic graphs G with δ(G) ≤ 1. Besides, we prove some inequalities involving the hyperbolicity constant and other parameters for cubic graphs.     

The Universal Functorial Equivariant Lefschetz Invariant

We introduce the universal functorial equivariant Lefschetz invariant for endomorphisms of finite proper G-CW-complexes, where G is a discrete group. We use K₀ of the category of “ ϕ -endomorphisms of finitely generated free RΠ(G, X)-modules”. We derive results about fixed points of equivariant endomorphisms of cocompact proper smooth G-manifolds. Received: February 2006     

Generalization of a theorem of Gonchar

Let X and Y be two complex manifolds, let D⊂X and G⊂Y be two nonempty open sets, let A (resp. B) be an open subset of ∂D (resp. ∂G), and let W be the 2-fold cross ((D∪A)×B)∪(A×(B∪G)). Under a geometric condition on the boundary sets A and B, we show that every function locally bounded, separately continuous on W, continuous on A×B, and separately holomorphic on (A×G)∪(D×B) “extends” to a function continuous on a “domain of holomorphy” and holomorphic on the interior of .     

Restricted Mean Value Property for Balayage Spaces with Jumps

It is shown that, for α-stable processes (Riesz potentials) or—more generally—for balayage spaces with jumps, “one-radius” results for harmonicity can be obtained under fairly weak assumptions.     

Strong rigidity of II1 factors arising from malleable actions of w-rigid groups, II

We prove that any isomorphism θ:M₀≃M of group measure space II₁ factors, , , with G₀ an ICC group containing an infinite normal subgroup with the relative property (T) of Kazhdan-Margulis (i.e. G₀w-rigid) and σ a Bernoulli action of some ICC group G, essentially comes from an isomorphism of probability spaces which conjugates the actions with respect to some identification G₀≃G. Moreover, any isomorphism θ of M₀ onto a “corner” pMp of M, for p∈M an idempotent, forces p=1. In particular, all group measure space factors associated with Bernoulli actions of w-rigid ICC groups have trivial fundamental group and any isomorphism of such factors comes from an isomorphism of the corresponding groups. This settles a “group measure space version” of Connes rigidity conjecture, shown in fact to hold true in a greater generality than just for ICC property (T) groups. We apply these results to ergodic theory, establishing new strong rigidity and superrigidity results for orbit equivalence relations.     

On minimal,strongly proximal actions of locally compact groups

Minimal, strongly proximal actions of locally compact groups on compact spaces, also known asboundary actions, were introduced by Furstenberg in the study of Lie groups. In particular, the action of a semi-simple real Lie groupG on homogeneous spacesG/Q, whereQ ⊂G is a parabolic subgroup, are boundary actions. Countable discrete groups admit a wide variety of boundary actions. In this note we show that ifX is a compact manifold with a faithful boundary action of some locally compact groupH, then (under some mild regularity assumption) the groupH, the spaceX, and the action split into a direct product of a semi-simple Lie groupG acting onG/Q and a boundary action of a discrete countable group. The author was partially supported by NSF grants DMS-0049069, 0094245 and GIF grant G-454-213.06/95.     

Isometry Groups of Proper Hyperbolic Spaces

Let X be a proper hyperbolic geodesic metric space and let G be a closed subgroup of the isometry group Iso(X) of X. We show that if G is not elementary then for every p ∈ (1, ∞) the second continuous bounded cohomology group H²_cb(G, L^p(G)) does not vanish. As an application, we derive some structure results for closed subgroups of Iso(X). Partially supported by Sonderforschungsbereich 611.     

Dichotomies of the set of test measures of a Haar-null set

We prove that ifX is a Polish space andF a face ofP(X) with the Baire property, thenF is either a meager or a co-meager subset ofP(X). As a consequence we show that for every abelian Polish groupX and every analytic Haar-null set Λ⊆X, the set of test measuresT(Λ) of Λ is either meager or co-meager. We characterize the non-locally-compact groups as the ones for which there exists a closed Haar-null setF⊆X withT(F) meager, Moreover, we answer negatively a question of J. Mycielski by showing that for every non-locally-compact abelian Polish group and every σ-compact subgroupG ofX there exists aG-invariantF _σ subset ofX which is neither prevalent nor Haar-null. Research supported by a grant of EPEAEK program “Pythagoras”.     

Mader’s Conjecture On Extremely Critical Graphs

A non-complete graph G is called an (n,k)-graph if it is n-connected but G—X is not (n−|X|+1)-connected for any X ⊂V (G) with |X|≤k. Mader conjectured that for k≥3 the graph K_2k+2−(1−factor) is the unique (2k,k)-graph(up to isomorphism). Here we prove this conjecture.     

Vapnik-Chervonenkis dimension and (pseudo-)hyperplane arrangements

An arrangement of oriented pseudohyperplanes in affined-space defines on its setX of pseudohyperplanes a set system (or range space) (X, ℛ), ℛ ⊑ 2 ^x of VC-dimensiond in a natural way: to every cellc in the arrangement assign the subset of pseudohyperplanes havingc on their positive side, and let ℛ be the collection of all these subsets. We investigate and characterize the range spaces corresponding tosimple arrangements of pseudohyperplanes in this way; such range spaces are calledpseudogeometric, and they have the property that the cardinality of ℛ is maximum for the given VC-dimension. In general, such range spaces are calledmaximum, and we show that the number of rangesR∈ℛ for whichX - R∈ℛ also, determines whether a maximum range space is pseudogeometric. Two other characterizations go via a simple duality concept and “small” subspaces. The correspondence to arrangements is obtained indirectly via a new characterization of uniforom oriented matroids: a range space (X, ℛ) naturally corresponds to a uniform oriented matroid of rank |X|—d if and only if its VC-dimension isd,R∈ℛ impliesX - R∈ℛ, and |ℛ| is maximum under these conditions. Part of this work was done while the first author was a member of the Graduiertenkolleg “Algorithmische Diskrete Mathematik,” supported by the Deutsche Forschungsgemeinschaft, Grant We 1265/2-1. Part of this work has been supported by the German-Israeli Foundation for Scientific Research and Development (G.I.F.).     

Quasiperiodic spectra and orthogonality for iterated function system measures

We extend classical basis constructions from Fourier analysis to attractors for affine iterated function systems (IFSs). This is of interest since these attractors have fractal features, e.g., measures with fractal scaling dimension. Moreover, the spectrum is then typically quasi-periodic, but non-periodic, i.e., the spectrum is a “small perturbation” of a lattice. Due to earlier research on IFSs, there are known results on certain classes of spectral duality-pairs, also called spectral pairs or spectral measures. It is known that some duality pairs are associated with complex Hadamard matrices. However, not all IFSs X admit spectral duality. When X is given, we identify geometric conditions on X for the existence of a Fourier spectrum, serving as the second part in a spectral pair. We show how these spectral pairs compose, and we characterize the decompositions in terms of atoms. The decompositions refer to tensor product factorizations for associated complex Hadamard matrices. Research supported in part by a grant from the National Science Foundation DMS-0704191.