Groups with infinitely many types of fixed subgroups

It is a theorem of Shor that ifG is a word-hyperbolic group, then up to isomrphism, only finitely many groups appear as fixed subgroups of automorphisms ofG. We give an example of a groupG acting freely and cocompactly on a CAT(0) square complex such that infinitely many non-isomorphic groups appear as fixed subgroups of automorphisms ofG. Consequently, Shor’s finiteness result does not hold if the negative curvature condition is relaxed to either biautomaticity or nonpositive curvature. D. T. Wise was supported by grants from FCAR and NSERC.     

Collapsing and Dirac-Type Operators

We analyze the limit of the spectrum of a geometric Dirac-type operator under a collapse with bounded diameter and bounded sectional curvature. In the case of a smooth limit space B, we show that the limit of the spectrum is given by the spectrum of a certain first-order differential operator on B, which can be constructed using superconnections. In the case of a general limit space X, we express the limit operator in terms of a transversally elliptic operator on a G-manifold X/ with X = X//G. As an application, we give a characterization of manifolds which do not admit uniform upper bounds, in terms of diameter and sectional curvature, on the k-th eigenvalue of the square of a Dirac-type operator. We also give a formula for the essential spectrum of a Dirac-type operator on a finite-volume manifold with pinched negative sectional curvature.     

Coupling vs. conductance for the Jerrum–Sinclair chain*

We address the following question: is the causal coupling method as strong as the conductance method in showing rapid mixing of Markov chains? A causal coupling is a coupling which uses only past and present information, but not information about the future. We answer the above question in the negative by showing that there exists a bipartite graph G such that any causal coupling argument on the Jerrum–Sinclair Markov chain for sampling almost uniformly from the set of perfect and near perfect matchings of G must necessarily take time exponential in the number of vertices in G. In contrast, the above Markov chain on G has been shown to mix in polynomial time using conductance arguments. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 18: 1–17, 2001     

New upper bounds for the chromatic number of a graph*

We show that for any graph G, the chromatic number χ(G) ≤ Δ₂(G) + 1, where Δ₂(G) is the largest degree that a vertex ν can have subject to the condition that ν is adjacent to a vertex whose degree is at least as big as its own. Moreover, we show that the upper bound is best possible in the the following sense: If Δ₂(G) ≥ 3, then to determine whether χ(G) ≤ Δ₂(G) is an NP‐complete problem. © 2001 John Wiley & Sons, Inc. J Graph Theory 36: 117–120, 2001     

Estimates for Derivatives of the Green Functions for Noncoercive Differential Operators on Homogeneous Manifolds of Negative Curvature

We consider the Green functions G for second-order noncoercive differential operators on homogeneous manifolds of negative curvature, being a semi-direct product of a nilpotent Lie group N and A=R ⁺. Using some probabilistic and analytic techniques we obtain estimates for derivatives of the Green functions G with respect to the N and A-variables.     

Holomorphic two-spheres in complex Grassmann manifold <Emphasis Type="Italic">G</Emphasis>(2, 4)

In this paper, we use the harmonic sequence to study the linearly full holomorphic two-spheres in complex Grassmann manifold G(2, 4). We show that if the Gaussian curvature K (with respect to the induced metric) of a non-degenerate holomorphic two-sphere satisfies K ≤ 2 (or K ≥ 2), then K must be equal to 2. Simultaneously, we show that one class of the holomorphic two-spheres with constant curvature 2 is totally geodesic. Concerning the degenerate holomorphic two-spheres, if its Gaussian curvature K ≤ 1 (or K ≥ 1), then K = 1. Moreover, we prove that all holomorphic two-spheres with constant curvature 1 in G(2, 4) must be U (4)-equivalent.     

Cohomogeneity one Riemannian manifolds of non-positive curvature

We study a G-manifold M which admits a G-invariant Riemannian metric g of non-positive curvature. We describe all such Riemannian G-manifolds (M,g) of non-positive curvature with a semisimple Lie group G which acts on M regularly and classify cohomogeneity one G-manifolds M of a semisimple Lie group G which admit an invariant metric of non-positive curvature. Some results on non-existence of invariant metric of negative curvature on cohomogeneity one G-manifolds of a semisimple Lie group G are given.     

Solution of a Busemann problem

We give a solution to the problem posed by Busemann which is as follows: Determine the noncompact Busemann G-spaces such that for every two geodesics there exists a motion taking one to the other. We prove that each of these spaces is isometric to the Euclidean space or to one of the noncompact symmetric spaces of rank 1 (of negative sectional curvature).     

On an invariant on isometric immersions into spaces of constant sectional curvature

In the present paper, we give an invariant on isometric immersions into spaces of constant sectional curvature. This invariant is a direct consequence of the Gauss equation and the Codazzi equation of isometric immersions. We apply this invariant on some examples. Further, we apply it to codimension 1 local isometric immersions of 2-step nilpotent Lie groups with arbitrary leftinvariant Riemannian metric into spaces of constant nonpositive sectional curvature. We also consider the more general class, namely, three-dimensional Lie groups G with nontrivial center and with arbitrary left-invariant metric. We show that if the metric of G is not symmetric, then there are no local isometric immersions of G into Q _c ⁴.     

The geodesic flow of a nonpositively curved graph manifold

We consider discrete cocompact isometric actions where X is a locally compact Hadamard space (following [B] we will refer to CAT(0) spaces — complete, simply connected length spaces with nonpositive curvature in the sense of Alexandrov — as Hadamard spaces) and G belongs to a class of groups (“admissible groups”) which includes fundamental groups of 3-dimensional graph manifolds. We identify invariants (“geometric data”) of the action which determine, and are determined by, the equivariant homeomorphism type of the action of G on the ideal boundary of X. Moreover, if are two actions with the same geometric data and is a G-equivariant quasi-isometry, then for every geodesic ray there is a geodesic ray (unique up to equivalence) so that . This work was inspired by (and answers) a question of Gromov in [Gr3, p. 136]. Submitted: May 2001.     

On holomorphic principal bundles over a compact Riemann surface admitting a flat connection

Let G be a connected reductive linear algebraic group over , and X a compact connected Riemann surface. Let be a Levi factor of some parabolic subgroup of G, with its maximal abelian quotient. We prove that a holomorphic G-bundle over X admits a flat connection if and only if for every such L and every reduction of the structure group of to L, the -bundle obtained by extending the structure group of is topologically trivial. For , this is a well-known result of A. Weil. Received: 1 December 2000 / Revised version: 2 April 2001 / Published online: 24 September 2001     

Pontryagin duality for topological Abelian groups

A topological Abelian group G is Pontryagin reflexive, or P-reflexive for short, if the natural homomorphism of G to its bidual group is a topological isomorphism. We look at the question, set by Kaplan in 1948, of characterizing the topological Abelian groups that are P-reflexive. Thus, we find some conditions on an arbitrary group G that are equivalent to the P-reflexivity of G and give an example that corrects a wrong statement appearing in previously existent characterizations of P-reflexive groups. Received: 10 February 2000 / Published online: 17 May 2001     

Linear embeddings of semialgebraic G-spaces

Let G be a compact semialgebraic linear group. We prove that every regular semialgebraic G-space admits a semialgebraic G-embedding into some semialgebraic orthogonal representation space of G. Received: 9 January 2001; in final form: 17 August 2001 / Published online: 28 February 2002     

Homogeneous Metrics with Nonnegative Curvature

Given compact Lie groups H⊂G, we study the space of G-invariant metrics on G/H with nonnegative sectional curvature. For an intermediate subgroup K between H and G, we derive conditions under which enlarging the Lie algebra of K maintains nonnegative curvature on G/H. Such an enlarging is possible if (K,H) is a symmetric pair, which yields many new examples of nonnegatively curved homogeneous metrics. We provide other examples of spaces G/H with unexpectedly large families of nonnegatively curved homogeneous metrics.     

Acyclic edge colorings of graphs

A proper coloring of the edges of a graph G is called acyclic if there is no 2‐colored cycle in G. The acyclic edge chromatic number of G, denoted by a′(G), is the least number of colors in an acyclic edge coloring of G. For certain graphs G, a′(G) ≥ Δ(G) + 2 where Δ(G) is the maximum degree in G. It is known that a′(G) ≤ 16 Δ(G) for any graph G. We prove that there exists a constant c such that a′(G) ≤ Δ(G) + 2 for any graph G whose girth is at least cΔ(G) log Δ(G), and conjecture that this upper bound for a′(G) holds for all graphs G. We also show that a′(G) ≤ Δ + 2 for almost all Δ‐regular graphs. © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 157–167, 2001     

Kazhdan's property (T), L^2-spectrum and isoperimetric inequalities for locally symmetric spaces

Let V = G\G/KV =\Gamma\backslash G/K be a Riemannian locally symmetric space of nonpositive sectional curvature and such that the isometry group G of its universal covering space has Kazhdan's property (T). We establish strong dichotomies between the finite and infinite volume case. In particular, we characterize lattices (or, equivalently, arithmetic groups) among discrete subgroups G ì G\Gamma\subset G in various ways (e.g., in terms of critical exponents, the bottom of the spectrum of the Laplacian and the behaviour of the Brownian motion on V).     

The relation between the Baum-Connes Conjecture and the Trace Conjecture

We prove a version of the L ²-index Theorem of Atiyah, which uses the universal center-valued trace instead of the standard trace. We construct for G-equivariant K-homology an equivariant Chern character, which is an isomorphism and lives over the ring ℤ⊂λ ^G ⊂ℚ obtained from the integers by inverting the orders of all finite subgroups of G. We use these two results to show that the Baum-Connes Conjecture implies the modified Trace Conjecture, which says that the image of the standard trace K ₀(C ^* _r (G))→ℝ takes values in λ ^G . The original Trace Conjecture predicted that its image lies in the additive subgroup of ℝ generated by the inverses of all the orders of the finite subgroups of G, and has been disproved by Roy [15]. Oblatum 10-IV-2001 & 18-X-2001?Published online: 15 April 2002     

Nonnegatively curved fixed point homogeneous 5-manifolds

Let G be a compact Lie group acting effectively by isometries on a compact Riemannian manifold M with nonempty fixed point set Fix(M, G). We say that the action is fixed point homogeneous if G acts transitively on a normal sphere to some component of Fix(M, G), equivalently, if Fix(M, G) has codimension one in the orbit space of the action. We classify up to diffeomorphism closed, simply connected 5-manifolds with nonnegative sectional curvature and an effective fixed point homogeneous isometric action of a compact Lie group.     

The Poisson Boundary of Amenable Extensions

 We obtain two new criteria of amenability of a closed normal subgroup H in a locally compact group G. The first one is formulated in terms of convergence of convolutions of probability measures on G, and the second one in terms of the Poisson boundaries of random walks on G and on the quotient group G/H. Received 10 April 2001; in revised form 18 December 2001     

Representations of Association Schemes and Their Factor Schemes

 In the present paper we investigate the relationship between the complex representations of an association scheme G and the complex representations of certain factor schemes of G. Our first result is that, similar to group representation theory, representations of factor schemes over normal closed subsets of G can be viewed as representations of G itself. We then give necessary and sufficient conditions for an irreducible character of G to be a character of a factor scheme of G. These characterizations involve the central primitive idempotents of the adjacency algebra of G and they are obtained with the help of the Frobenius reciprocity low which we prove for complex adjacency algebras. Received: February 27, 2001 Final version received: August 30, 2001