arXiv research

New invariant real rank identifies constant real Lie algebroids.

Let G be a connected semisimple Lie group without compact factors whose real rank is at least 2, and let Γ\subset G be an irreducible lattice. We provide a C^\infty classification for volume-preserving Cartan actions of Γand G. Also, if G has real rank at least 3, we provide a C^\infty classification for volume-preserv…

If Gamma is a nonuniform, irreducible lattice in a semisimple Lie group whose real rank is greater than 1, we show Gamma contains a subgroup that is isomorphic to a nonuniform, irreducible lattice in either SL(3,R), SL(3,C), or a direct product SL(2,R)^m x SL(2,C)^n$, with m + n > 1. (In geometric terms, this can be in…

Study finds lower bounds on flat cycles in congruence covers of symmetric spaces.

Study bounds on lattice actions on compact pseudo-Riemannian manifolds.

Study shows nonexistence of certain geometric structures in complex geometries.

Let $G$ and $G'$ be simple Lie groups of equal real rank and real rank at least $2$. Let $Γ<G$ and $Λ< G'$ be non-uniform lattices. We prove a theorem that often implies that any quasi-isometric embedding of $Γ$ into $Λ$ is at bounded distance from a homomorphism. For example, any quasi-isometric embedding of $SL(n,\ma…

Study of representation equivalence and isospectrality in symmetric spaces of real rank one.

Study growth of systoles in arithmetic manifolds, focusing on $k$-dimensional cases.

Given a simple Lie group $H$ of real rank at least $2$ we show that the maximum cardinality of a set of isospectral non-isometric $H$-locally symmetric spaces of volume at most $x$ grows at least as fast as $x^{c\log x/ (\log\log x)^2}$ where $c = c(H)$ is a positive constant. In contrast with the real rank $1$ case, t…

We consider actions of non-compact simple Lie groups preserving an analytic rigid geometric structure of algebraic type on a compact manifold. The structure is not assumed to be unimodular, so an invariant measure may not exist. Ergodic stationary measures always exist, and when such a measure has full support, we show…

Given a simple Lie group G of rank 1, we consider compact pseudo-Riemannian manifolds (M,g) of signature (p,q) on which G can act conformally. Precisely, we determine the smallest possible value for the index min(p,q) of the metric. When the index is optimal and G non-exceptional, we prove that the metric must be confo…

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 maximal flats of the symmetric space of G. We define its Chabauty compactification…

If $G$ is a semisimple Lie group of real rank at least 2 and $Γ$ is an irreducible lattice in $G$, then every homomorphism from $Γ$ to the outer automorphism group of a finitely generated free group has finite image.

Let H be a closed, connected subgroup of a connected, simple Lie group G with finite center. The homogeneous space G/H has a "tessellation" if there is a discrete subgroup D of G, such that D acts properly discontinuously on G/H, and the double-coset space D\G/H is compact. Note that if either H or G/H is compact, then…

Higher index theorem for Dirac operators on finite-volume spaces.

Using conjugation of Shimura varieties, we produce nonisomorphic, cocompact, torsion-free lattices in $\mathrm{PU}(n,1)$ with isomorphic profinite completions for all $n \ge 2$. This disproves a conjecture of D. Kazhdan and gives the first examples nonisomorphic lattices in a semisimple Lie group of real rank one with …

We prove the finiteness of the cohomology of torsion-free lattices in a semisimple Lie group of real rank one with coefficients in the distribution vector globalization of Harish-Chandra modules. The cohomology is expressed in terms of automorphic and cusp forms. We also consider the Lie-algebra cohomology of these glo…

Improved homological dimension for certain subgroups in Lie groups.

We introduce the notion of CR quaternionic map and we prove that any such real-analytic map, between CR quaternionic manifolds, is the restriction of a quaternionic map between quaternionic manifolds. As an application, we prove, for example, that for any submanifold $M$, of dimension $4k-1$, of a quaternionic manifold…

If f is a conformal mapping defined on a connected open subset of a Carnot group G, then either f is the composition of a translation, a dilation and an isometry, or G is the nilpotent Iwasawa component of a real rank 1 simple Lie group S, and f arises from the action of S on G, viewed as an open subset of S/P, where P…

We obtain the Plancherel theorem for the quotient of a simple Lie group of real rank one by a convex-cocompact discrete subgroup and its consequences for the spectrum of locally invariant differential operators on bundles over Kleinian manifolds. We develop a geometric version of scattering theory. The paper is an upda…

We show that codimension one dimensional Jacobian of the barycentric straightening map is uniformly bounded for most of the higher rank symmetric spaces. As a consequence, we prove that the locally finite simplicial volume of most $\mathbb Q$-rank $1$ locally symmetric spaces is positive, which has been open for many y…

We obtain restrictions on the topology of a closed connected manifold B that bounds a (possibly noncompact) manifold whose interior V admits a complete Riemannian metric of nonpositive sectional curvature. If G denotes the fundamental group of B, then a sample result is that B must be aspherical and incompressible if o…

This is the geometric part of two papers on the cohomology of Kaehler groups. Using non-Abelian Hodge theory we show that if a finitely presented group with an unbounded complex linear morphism is the fundamental group of a compact Kaehler manifold then its second or its fourth Betti number does not vanish. Combined wi…

Proof of boundedness of quasimorphisms for certain Lie groups.

The paper explores higher property T in lattices and its connections to geometric phenomena.

Let G be a semisimple Lie group with no compact factors, K a maximal compact subgroup of G, and $Γ$ a lattice in G. We study automorphic forms for $Γ$ if G is of real rank one with some additional assumptions, using dynamical approach based on properties of the homogeneous flow on $Γ\backslash G$ and a Livshitz type th…

Let G be a finitely generated group having the property that any action of any finite-index subgroup of G by homeomorphisms of the circle must have a finite orbit. (By a theorem of E.Ghys, lattices in simple Lie groups of real rank at least two have this property.) Suppose that such a G acts on a compact manifold M by …

Lie foliations with symmetric leaves are smoothly conjugate to homogeneous ones.

Formula for Lefschetz numbers on locally symmetric spaces.

Let G be a real semisimple Lie group with no compact factors and finite centre, and let $Λ$ be a lattice in G. Suppose that there exists a homomorphism from $Λ$ to the outer automorphism group of a right-angled Artin group $A_Γ$ with infinite image. We give an upper bound to the real rank of G that is determined by the…

For convex domains with $C^{1,ε}$ boundary we give a precise description of the automorphism group: if an orbit of the automorphism group accumulates on at least two different closed complex faces of the boundary, then the automorphism group has finitely many components and the connected component of the identity is th…

We consider conformal actions of simple Lie groups on compact Lorentzian manifolds. Mainly motivated by the Lorentzian version of a conjecture of Lichnerowicz, we establish the alternative: Either the group acts isometrically for some metric in the conformal class, or the manifold is conformally flat - that is, everywh…

We derive a formula for the regularized trace of operators with compact spectrum which act on the space of square integrable functions on the quotient of a semisimple Liegroup of real rank one by a convex-cocompact subgroup. The sum of normalized orbital integrals associated to the hyperbolic conjugacy classes of this …

We study properly discontinuous and cocompact actions of a discrete subgroup $Γ$ of an algebraic group $G$ on a contractible algebraic manifold $X$. We suppose that this action comes from an algebraic action of $G$ on $X$ such that a maximal reductive subgroup of $G$ fixes a point. When the real rank of any simple subg…

Let $J$ be a semisimple Lie group with all simple factors of real rank at least two. Let $Γ<J$ be a lattice. We prove a very general local rigidity result about actions of $J$ or $Γ$. This shows that almost all so-called "standard actions" are locally rigid. As a special case, we see that any action of $Γ$ by toral aut…

This book provides a gentle introduction to the study of arithmetic subgroups of semisimple Lie groups. This means that the goal is to understand the group SL(n,Z) and certain of its subgroups. Among the major results discussed in the later chapters are the Mostow Rigidity Theorem, the Margulis Superrigidity Theorem, R…

Study on manifolds with higher-rank lattice actions in projective and conformal classes.

We give a dynamical description, in terms of a Weil-type zeta function, to the holomorphic torsion with coefficients for certain compact Hermitian locally symmetric manifolds, whose connected group G of isometries of the universal cover has only one conjugacy class of cuspidal maximal parabolic subgroup and satisfies a…

Let G be a simple Lie group of real rank one, and S the ideal boundary of the corresponding symmetric space of noncompact type (H^n_R, H^n_C, H^n_H or H^2_O). We show the finiteness of the possible values of the secondary characteristic classes of transversely homogeneous foliations on a fixed manifold whose transverse…

In the first paper of this series (arxiv.org/abs/1210.2961) we studied the asymptotic behavior of Betti numbers, twisted torsion and other spectral invariants for sequences of lattices in Lie groups G. A key element of our work was the study of invariant random subgroups (IRSs) of G. Any sequence of lattices has a subs…

Let D be an irreducible lattice in a connected, semisimple Lie group G with finite center. Assume that the real rank of G is at least two, that G/D is not compact, and that G has more than one noncompact simple factor. We show that D has no orientation-preserving actions on the real line. (In algebraic terms, this mean…

Let L be a Lie group and Lambda a lattice in L. Suppose G is a non-compact simple Lie group realized as a Lie subgroup of L, and the image of G on L/Lambda is dense. Let c be a diagonalizable element of G not contained in a compact subgroup. Let U be the expanding horospherical subgroup of G associated to c. Let Omega …

Paper finds surface groups can deform in reductive symmetric spaces.

A canonically defined mod 2 linear dependency current is associated to each collection of m sections of a real rank n vector bundle. This current is supported on the linear dependency set of the collection of sections. It is defined whenever the collection satisfies a weak measure theoretic condition called "atomicity"…

Let $G$ be a real simple linear connected Lie group of real rank one. Then, $X := G/K$ is a Riemannian symmetric space with strictly negative sectional curvature. By the classification of these spaces, $X$ is a real/complex/quaternionic hyperbolic space or the Cayley hyperbolic plane. We define the Schwartz space $…

We study the asymptotic behaviour of Betti numbers, twisted torsion and other spectral invariants of sequences of locally symmetric spaces. Our main results are uniform versions of the DeGeorge--Wallach Theorem, of a theorem of Delorme and various other limit multiplicity theorems. A basic idea is to adapt the notion o…