Cocompact Lattice Research Articles

Nonstationary normal forms and rigidity of group actions

We develop a proper “nonstationary” generalization of the classical theory of normal forms for local contractions. In particular, it is shown under some assumptions that the centralizer of a contraction in an extension is a particular Lie group, determined by the spectrum of the linear part of the contractions. We show that most homogeneous Anosov actions of higher rank abelian groups are locally C∞ rigid (up to an automorphism). This result is the main part in the proof of local C∞ rigidity for two very different types of algebraic actions of irreducible lattices in higher rank semisimple Lie groups: (i) the actions of cocompact lattices on Furstenberg boundaries, in particular projective spaces, and (ii) the actions by automorphisms of tori and nilmanifolds. The main new technical ingredient in the proofs is the centralizer result

Vanishing theorems for cohomologies of automorphic vector bundles

Let M be a compact, irreducible, locally hermitian symmetric space of noncompact type. If M is not a Riemann surface, then Calabi and Vesentini have shown in ([4]), that the complex structure on M is infinitesimally rigid, i.e., they show the vanishing of H(M, ΘM), where ΘM is the tangent sheaf of germs of holomorphic vector fields on M. Their method involves the construction of an 'auxiliary expression', which is simplified in two different ways, to obtain a quadratic form involving curvature terms. The desired vanishing is reduced then to proving that the quadratic form is positive definite. One obtains criteria for the vanishing of the cohomology groups H*(M, ΘM), which depend on the curvature properties of M and not on the lattice defining M. Based on their method, Weil showed (see [17]) that an irreducible, cocompact lattice Γ in a real semisimple Lie group G without compact or three dimensional factors is rigid, i.e., any 'nearby' deformations of Γ inside G are equivalent. This amounts to showing the vanishing of ΐί (Γ, Ad), where Ad is the adjoint representation of G on its Lie algebra. Matsushima refined this method to show vanishing of Betti numbers of M below some degree ([10]). Our main result is to give a criterion for the vanishing of cohomologies of vector bundles, generalising the results of Calabi-Vesentini and Matsushima. More generally we give a criterion for the vanishing of d-cohomology (or (g, K)— cohomology) of unitary gmodules with coeffecients in a Kmodule. The terms involved can be calculated explicitly in terms of the dominant weight of the reprsentation defining the automorphic vector bundle and the curva-

Torsion in cocompact lattices in coverings of spin (2,n)

Torsion in cocompact lattices in coverings of spin (2,n)

Deformations of complex structures on Γ/SL 2(C)

LetG be a connected complex semisimple Lie group. Let Γ be a cocompact lattice inG. In this paper, we show that whenG isSL 2(C), nontrivial deformations of the canonical complex structure onX exist if and only if the first Betti number of the lattice Γ is non-zero. It may be remarked that for a wide class of arithmetic groups Γ, one can find a subgroup Γ′ of finite index in Γ, such that Γ′/[Γ′,Γ′] is finite (it is a conjecture of Thurston that this is true for all cocompact lattices inSL(2, C)).

Isoperimetric inequalities for the fundamental groups of torus bundles over the circle

We give upper bounds for isoperimetric functions of semi-direct products in terms of the asymptotic behaviour of ||Ak|| ask → ∞. In the caseA ∈ Sp(n, ℤ) we show that these bounds are sharp. This enables us to describe infinite families of nilpotent groups whose Dehn functions are bounded above and below by polynomials of degree the nilpotency class plus 1. We also recover the isoperimetric inequalities of cocompact lattices inNil andSol.

Discrete groups and non-Riemannian homogeneous spaces

A basic question in geometry is to understand compact locally homogeneous manifolds, i.e., those compact manifolds that can be locally modelled on a homogeneous space J\H of a finite-dimensional Lie group H. This means that there is an atlas on a manifold M consisting of local diffeomorphisms with open sets in J\H where the transition functions between these open sets are given by translations by elements of H. A basic example is the case of M = J\H/F where F c H is a discrete subgroup acting freely and properly discontinuously on J/H, with a compact quotient. (These examples can be abstractly characterized as the case. See [G].) We then call F a cocompact lattice on J\H. When J is compact, J\H has an H-invariant Riemannian metric, so that M itself becomes a Riemannian manifold in a natural way. Thus, the class of locally homogeneous Riemannian manifolds includes (but is not exhausted by) the locally symmetric spaces. For H Aff(Rn) and J = GL(n, R), the spaces J\H/F are the complete affinely flat manifolds, a non-Riemannian example which has been extensively studied. When H is simple and J is not compact, the situation is far from being well understood. Some special series of such homogeneous spaces admitting a cocompact lattice have been constructed by Kulkarni [Ku] and by T. Kobayashi [K]. For example, Kulkarni shows the existence of such a discrete group F for the homogeneous spaces SO(1, SO(2, 2n) and Kobayshi does the same for U( 1, n) SO(2, 2n). On the other hand, the constructions they give for these (and other similar) series are quite special, and there has been the general suspicion that most homogeneous spaces J\H with H simple and J not compact do not admit a cocompact lattice. (We recall that, when J is compact, i.e. the Riemannian case, such F always exist by a result of Borel [B].) One general situation in which it is known that no such F exists is the case in which J\H is noncompact and of reductive type and R-rank(H) = R-rank(J). This situation was studied by Calabi-Markus [CM], Wolf [W], and Kobayashi [K], and the nonexistence of F in this situation is a special case of a more general result known as the Calabi-Markus phenomenon. Kobayashi and Ono [KO] use the Hirzebruch proportionality principle to show that some other special series do

The Poisson boundary for rank one manifolds and their cocompact lattices

The Poisson boundary for rank one manifolds and their cocompact lattices

Convex real projective structures on closed surfaces are closed

The deformation space C ( Σ ) \mathfrak {C}(\Sigma ) of convex R P 2 \mathbb {R}{{\mathbf {P}}^2} -structures on a closed surface Σ \Sigma with χ ( Σ ) > 0 \chi (\Sigma ) > 0 is closed in the space Hom ⁡ ( π , SL ⁡ ( 3 , R ) ) / SL ⁡ ( 3 , R ) \operatorname {Hom} (\pi ,\operatorname {SL} (3,\mathbb {R}))/\operatorname {SL} (3,\mathbb {R}) of equivalence classes of representations π 1 ( Σ ) → SL ⁡ ( 3 , R ) {\pi _1}(\Sigma ) \to \operatorname {SL} (3,\mathbb {R}) . Using this fact, we prove Hitchin’s conjecture that the contractible "Teichmüller component" (Lie groups and Teichmüller space, preprint) of Hom ⁡ ( π , SL ⁡ ( 3 , R ) ) / SL ⁡ ( 3 , R ) \operatorname {Hom} (\pi ,\operatorname {SL} (3,\mathbb {R}))/\operatorname {SL} (3,\mathbb {R}) precisely equals C ( Σ ) \mathfrak {C}(\Sigma ) .

Uniformly distributed orbits of certain flows on homogeneous spaces

is non-compact. For a co-compact lattice, this result was strengthened by F/irstenberg [1 lJ proving that every orbit is uniformly distributed with respect to a G-invariant measure. For non-uniform lattices in SL(2, R), using a classification of invariant measures obtained by Dani in [21, Dani and Smillie [31 proved that every non-periodic orbit is uniformly distributed. There are also various results obtained on orbit closures and invariant measures etc. of larger subgroups consisting of unipotent elements, especially the horospherical subgroups. Recently, there was a spurt in the area initiated by Margulis' proof (cf. [151, see also [7]) of Oppenheim conjecture on values of quadratic forms at integral points using the study of unipotent flows. The reader is referred to the survey articles by Dani [41 and Margulis [14J for various related developments. We now note some conjectures expected to hold for orbits of a unipotent flow, namely the U-action on

Lattices of Minimal Covolume in SL 2 : A Nonarchimedean Analogue of Siegel's Theorem μ≥π/21

A classical result of Siegel [Si] asserts that if G = SL2(R) and ,u is its Haar measure (appropriately normalized) then mintj,u(F\G) = 7r/21 , where r runs over all the discrete subgroups of G. The minimum is obtained for the triangle group (2, 3, 7) which is a cocompact lattice in G (see also [Gr]). For G = SL2(C), Meyerhoff [Me] has shown recently that among the nonuniform lattices in G (that is, discrete subgroups of finite covolume but not cocompact) the minimum volume is obtained with F = SL2(&3) where &3 is the ring of integers of Q(v/3), and it is approximately 0.0863. (But, there are cocompact lattices with smaller covolumes and the minimum is not known, see [EGM, Th]). In this paper, we address the analogue question for locally compact nonarchimedean fields. Our main result is: Let F be a locally compact field of characteristic p > 0 with residue field of order q = pa . (Hence F is isomorphic to Fq((I/t)), the field of Laurent formal power series in l/t ).

Unitary spectrum and the fundamental group for actions of semisimple Lie groups

Suppose a locally compact group G acts on a separable metrizable space M, preserving a finite measure #. Then there is an associated unitary representation of G on LZ(M,/~)o, the square integrable functions orthogonal to the constants. This representation will in general be far from irreducible, and in fact, when G is not compact need not have any irreducible subrepresentations. Thus, in the decomposition of LZ(G)o into irreducible constituents, one not only requires a direct integral rather than a direct sum, but the associated spectral measure on the unitary dual G (say for type I groups) will, except in special circumstances, have no atoms. The point of this paper is to give natural and general conditions, mostly involving just rq(M), under which there will be an infinite atomic spectrum. (I.e., the spectral measure has infinitely many atoms.) The basic example of actions in which the spectrum is purely atomic, i.e., when the representation is a direct sum of irreducible representations, is the case of M = G/F, where F c G is a cocompact discrete subgroup. In this case, assuming 9 G to be simply connected, we have 7tl(M ) = F. The first results of this paper show that when G is a simple Lie group with R-rank(G)=> 2, we can deduce that any action of G on a compact manifold M with fundamental group isomorphic to such a F, and satisfying a natural hypothesis of either a geometric or ergodic theoretic nature, must also have an infinite atomic spectrum. In fact, under certain natural hypotheses the same is true as long as there is a surjection of hi(M) onto F. To formulate this more precisely, let F c Ad(G) be a cocompact lattice, and let Pr be the unitary representation of Ad(G) on L2(Ad(G)/F)o . We recall that an action of a group on a manifold (or more generally on a standard space in the sense of [4]) is called engaging if there is no loss of ergodicity in passing to finite covers. (See I'4] for discussion.)

Deformation spaces associated to compact hyperbolic manifolds

In this paper we take a first step toward understanding representations of cocompact lattices in SO(n,1) into arbitrary Lie groups by studying the deformations of rational representations — see Proposition 5.1 for a rather general existence result. This proposition has a number of algebraic applications. For example, we remark that such deformations show that the Margulis Super-Rigidity Theorem, see [30], cannot be extended to the rank 1 case. We remark also that if Γ ⊂ SO(n,1) is one of the standard arithmetic examples described in Section 7 then Γ has a faithful representation ρ′ in SO(n+1), the Galois conjugate of the uniformization representation, and Proposition 5.1 may be used to deform the direct sum of ρ′ and the trivial representation in SO(n+2) thereby constructing non-trivial families of irreducible orthogonal representations of Γ. However, most of this paper is devoted to studying certain spaces of representations which are of interest in differential geometry in a sense which we now explain.

Torsion in cocompact lattices in coverings of spin (2,n)

Torsion in cocompact lattices in coverings of spin (2,n)

Co-compact lattices in complete Kac-Moody groups with Weyl group right-angled or a free product of spherical special subgroups

Let G be a complete Kac-Moody group of rank n \geq 2 over the finite field of order q, with Weyl group W and building \Delta. We first show that if W is right-angled, then for all q \neq 1 mod 4 the group G admits a cocompact lattice \Gamma which acts transitively on the chambers of \Delta. We also obtain a cocompact lattice for q =1 mod 4 in the case that \Delta is Bourdon's building. As a corollary of our constructions, for certain right-angled W and certain q, the lattice \Gamma has a surface subgroup. We also show that if W is a free product of spherical special subgroups, then for all q, the group G admits a cocompact lattice \Gamma with \Gamma a finitely generated free group. Our proofs use generalisations of our results in rank 2 concerning the action of certain finite subgroups of G on \Delta, together with covering theory for complexes of groups.

INTEGRALITY AND ARITHMETICITY OF CO-COMPACT LATTICE CORRESPONDING TO CERTAIN COMPLEX TWO-BALL QUOTIENTS OF PICARD NUMBER ONE

The main purpose of this paper is to provide some criteria for the integrality and arithmeticity of a cocompact lattice corresponding to a complex ball of dimension two.