Relative isoperimetric inequality and¶linear isoperimetric inequality for minimal submanifolds

We prove an optimal relative isoperimetric inequality

Foliations with leaves of nonpositive curvature

We answer a question of Gromov ([G2]) in the codimension 1 case: ifF is a codimension 1 foliation of a compact manifoldM with leaves of negative curvature, thenπ ₁(M) has exponential growth. We also prove a result analogous to Zimmer’s ([Z2]): ifF is a codimension 1 foliation on a compact manifold with leaves of nonpositive curvature, and ifπ ₁(M) has subexponential growth, then almost every leaf is flat. We give a foliated version of the Hopf theorem on surfaces without conjugate points. Partially supported by NSF Grant #DMS 9403870.     

Automorphisms of groups and of schemes of finite type

We show first that certain automorphism groups of algebraic varieties, and even schemes, are residually finite and virtually torsion free. (A group virtually has a property if some subgroup of finite index has it.) The rest of the paper is devoted to a study of the groups of automorphisms. Aut(Γ) and outer automorphisms Out(Γ) of a finitely generated group Γ, by using the finite-dimensional representations of Γ. This is an old idea (cf. the discussion of Magnus in [11]). In particular the classes of semi-simplen-dimensional representations of Γ are parametrized by an algebraic varietyS _n (Γ) on which Out(Γ) acts. We can apply the above results to this action and sometimes conclude that Out(Γ) is residually finite and virtually torsion free. This is true, for example, when Γ is a free group, or a surface group. In the latter case Out(Γ) is a “mapping class group.” Partially supported by the NSF under Grant MCS 80-05802.     

Cusp closing in rank one symmetric spaces

We investigate conditions under which cusps of pinched negative curvature can be closed as manifolds or orbifolds with nonpositive sectional curvature. We show that all cusps of complex hyperbolic type can be closed in this way whereas cusps of quaternionic or Cayley hyperbolic type cannot be closed. For cusps of real hyperbolic type we derive necessary and sufficient closing conditions. In this context we prove that a noncompact finite volume quotient of a rank one symmetric space can be approximated in the Gromov Hausdorff topology by closed orbifolds with nonpositive curvature if and only if it is real or complex hyperbolic. Using cusp closing methods we obtain new examples of real analytic manifolds of nonpositive sectional curvature and rank one containing flats. By the same methods we get an explicit resolution of the singularities in the Baily–Borel resp.Siu–Yau compactification of finite volume quotients of the complex hyperbolic space. Oblatum 2-IX-1994 & 7-VIII-1995     

On representations and <Emphasis Type="Italic">K</Emphasis>-theory of the braid groups

 Let Γ be the fundamental group of the complement of a K(Γ, 1) hyperplane arrangement (such as Artin's pure braid group) or more generally a homologically toroidal group as defined below. The triviality of bundles arising from orthogonal representations of Γ is characterized completely as follows. An orthogonal representation gives rise to a trivial bundle if and only if the representation factors through the spinor groups. Furthermore, the subgroup of elements in the complex K-theory of BΓ which arises from complex unitary representations of Γ is shown to be trivial. In the case of real K-theory, the subgroup of elements which arises from real orthogonal representations of Γ is an elementary abelian 2-group, which is characterized completely in terms of the first two Stiefel-Whitney classes of the representation. In addition, quadratic relations in the cohomology algebra of the pure braid groups which correspond precisely to the Jacobi identity for certain choices of Poisson algebras are shown to give the existence of certain homomorphisms from the pure braid group to generalized Heisenberg groups. These cohomology relations correspond to non-trivial Spin representations of the pure braid groups which give rise to trivial bundles. Received: 6 February 2002 / Revised version: 19 September 2002 / Published online: 8 April 2003 RID="⋆" ID="⋆" Partially supported by the NSF RID="⋆⋆" ID="⋆⋆" Partially supported by grant LEQSF(1999-02)-RD-A-01 from the Louisiana Board of Regents, and by grant MDA904-00-1-0038 from the National Security Agency RID="⋆" ID="⋆" Partially supported by the NSF Mathematics Subject Classification (2000): 20F36, 32S22, 55N15, 55R50     

Quadratic divergence of geodesics in CAT(0) spaces

A finite CAT(0) 2-complexX is produced whose universal cover possesses two geodesic rays which diverge quadratically and such that no pair of rays diverges faster than quadratically. This example shows that an aphorism in Riemannian geometry, that predicts that in nonpositive curvature nonasymptotic geodesic rays either diverge exponentially or diverge linearly, does not hold in the setting of CAT(0) complexes. The fundamental group ofX is that of a compact Riemannian manifold with totally geodesic boundary and nonpositive sectional curvature.Partially supported by NSF grant DMS-9200433     

Local rigidity for certain groups of toral automorphisms

Let Γ = SL(n, ℤ) or any subgroup of finite index, n ≥ 4. We show that the standard action of Γ on ⁿ is locally rigid, i.e., every action of Γ on ⁿ by C^∞ diffeomorphisms which is sufficiently close to the standard action is conjugate to the standard action by a C^∞ diffeomorphism. In the course of the proof, we obtain a global rigidity result (Theorem 4.12) for actions of free abelian subgroups of maximal rank in SL(n, ℤ). Partially supported by NSF grant DMS9011749.     

How often is 84(g−1) achieved?

For any finitely generated group Γ, the asymptotics of the set of orders of finite quotient groups of Γ are determined by the minimum dimension of a complex linear group containing an infinite quotient of Γ. We give a proof and an application to the asymptotic behavior of the set of integersg for which the Hurwitz bound is sharp. Partially supported by NSF Grant DMS-97-27553.     

On closures of orbits and arithmetic of quaternions

ForG=PGL₂(ℚ _p )×PGL₂ ℚ we study the closures of orbits under the maximal split Cartan subgroup ofG in homogeneous spacesΓ\G. We show that if a closure of an orbit contains a closed orbit then the orbit is either dense or closed. We show the relation of this to divisibility properties of integral quaternions and other lattices. Sponsored in part by the Edmund Landau Center for Research in Mathematical Analysis supported by the Minerva Foundation (Germany). Research at MSRI supported by NSF grant DMS8505550.     

Algebraic invariants for right-angled Artin groups

A finite simplicial graph Γ determines a right-angled Artin group G_Γ, with generators corresponding to the vertices of Γ, and with a relation υw=wυ for each pair of adjacent vertices. We compute the lower central series quotients, the Chen quotients, and the (first) resonance variety of G_Γ, directly from the graph Γ. Partially supported by NSF grant DMS-0311142.     

Structure of some ℤ-graded lie superalgebras of vector fields

In this paper we classifyℤ-graded transitive Lie superalgebras with prescribed nonpositive parts listed in [K2]. The classification of infinite-dimensional simple linearly compact Lie superalgebras given in [K2] is based on this result. We also study the structure of the exceptionalℤ-graded transitive Lie superalgebras and give their geometric realization. Dedicated to the memory of Claude Chevalley Partially supported by NSC grant 88-2115-M006-013 of the ROC Partially supported by NSF grant DMS-9622870 An erratum to this article is available at .     

Manifolds with quadratic curvature decay and fast volume growth

 We give sufficient conditions for a noncompact Riemannian manifold, which has quadratic curvature decay, to have finite topological type with ends that are cones over spherical space forms. Received: 21 March 2001 / Revised version: 21 March 2002 / Published online: 10 February 2003 Research partially supported by NSF grant DMS-0072154 and MSRI     

Curvature and uniformization

We approach the problem of uniformization of general Riemann surfaces through consideration of the curvature equation, and in particular the problem of constructing Poincaré metrics (i.e., complete metrics of constant negative curvature) by solving the equation Δu-e ^2u=K_o(z) on general open surfaces. A few other topics are discussed, including boundary behavior of the conformal factore ^2u giving the Poincaré metric when the Riemann surface has smoothly bounded compact closure, and also a curvature equation proof of Koebe's disk theorem. Research supported in part by NSF Grant DMS-9971975 and also at MSRI by NSF grant DMS-9701755. Research supported in part by NSF Grant DMS-9877077     

On locally convex hypersurfaces in Hadamard manifolds

Locally convex compact hypersurfaces immersed in a hollow simply connected Riemannian space of nonpositive sectional curvature are considered. They are proved to be convex hypersurfaces homeomorphic to the sphere. A similar result for immersed hypersurfaces with nonpositive definite second quadratic form of rank no smaller than one is obtained. Translated fromMatematicheskie Zametki, Vol. 67, No. 4, pp. 498–507, April, 2000.     

Corrigendum to “generalized martin’s axiom and souslin’s hypothesis for higher cardinals”

Correct proofs are given for Theorem 3 and the Propositions of §§5, 6 of [4]. For the latter, we must modify the principle (S)″ in a technical way. We prove a weaker version of Theorem 2, where □ is replaced by the stronger hypothesis PΓ_{N 1} ^b. Partially supported by NSF grant MCS 8301042.     

Weak mixing and unique ergodicity on homogeneous spaces

Under certain conditions the weak mixing of a translation onG/Γ implies that the action of an associated subgroup ofG onG/Γ is uniquely ergodic. This result generalizes earlier theorems of Furstenberg and Veech. Partially supported by the Sloan Foundation and NSF GP-14519.     

Actions of Cartan subgroups

Using affine Bruhat-Tits buildings, we associate certain subshifts of finite type to systems arising from the action of a Cartan subgroup of ap-adic semisimple Chevalley group on compact quotient spaces Γ\G. These are used to study the resulting dynamical systems. Sponsored in part by the Edmund Landau Center for Research in Mathematical Analysis supported by the Minerva Foundation (Federal Republic of Germany). Research at MSRI supported by NSF grant DMS8505550.     

Cusp closing in rank one symmetric spaces

We investigate conditions under which cusps of pinched negative curvature can be closed as manifolds or orbifolds with nonpositive sectional curvature. We show that all cusps of complex hyperbolic type can be closed in this way whereas cusps of quaternionic or Cayley hyperbolic type cannot be closed. For cusps of real hyperbolic type we derive necessary and sufficient closing conditions. In this context we prove that a noncompact finite volume quotient of a rank one symmetric space can be approximated in the Gromov Hausdorff topology by closed orbifolds with nonpositive curvature if and only if it is real or complex hyperbolic. Using cusp closing methods we obtain new examples of real analytic manifolds of nonpositive sectional curvature and rank one containing flats. By the same methods we get an explicit resolution of the singularities in the Baily-Borel resp. Siu-Yau compactification of finite volume quotients of the complex hyperbolic space.Oblatum 2-IX-1994 & 7-VIII-1995     

A theorem of H. Hopf and the Cauchy-Riemann inequality II

This is a sequel to “A theorem of H. Hopf and the Cauchy-Riemann inequality” [AdCT]. Here the result of the previous paper is extended (see the precise statement in Section 1 of the present paper) to surfaces in three-dimensional homogeneous Riemannian manifolds whose group of isometries has dimension four and the bundle curvature is nonzero, whereas in the previous paper only the case of vanishing bundle curvature was treated. * Partially supported by MEC-FEDER Grant No. MTM 2004-00160.     

On Einstein Hermitian manifolds

We show that every compact Einstein Hermitian surface with constant *–scalar curvature is a K?hler surface. In contrast to the 4-dimensional case, it is shown that there exists a compact Einstein Hermitian (4n + 2)-dimensional manifold with constant *–scalar curvature which is not K?hler. This study is supported by Kangwon National University.