K-groups associated with substitution minimal systems

Two ordered Bratteli diagrams can be constructed from an aperiodic substitution minimal dynamical system. One, the proper diagram, has a single maximal path and a single minimal path and the Vershik map on the path space can be extended homeomorphically to a map conjugate to the substitution system. The other, the improper diagram, encodes the substitution more naturally but often has many maximal and minimal paths and no continuous compact dynamics. This paper connects the two diagrams by considering theirK ₀-groups, obtaining the equation

Self-concordant barriers for hyperbolic means

The geometric mean and the function (det(·)) ^1/m (on the m-by-m positive definite matrices) are examples of “hyperbolic means”: functions of the form p ^1/m , where p is a hyperbolic polynomial of degree m. (A homogeneous polynomial p is “hyperbolic” with respect to a vector d if the polynomial t↦p(x+td) has only real roots for every vector x.) Any hyperbolic mean is positively homogeneous and concave (on a suitable domain): we present a self-concordant barrier for its hypograph, with barrier parameter O(m ²). Our approach is direct, and shows, for example, that the function −mlog(det(·)−1) is an m ²-self-concordant barrier on a natural domain. Such barriers suggest novel interior point approaches to convex programs involving hyperbolic means. Received: December 2, 1999 / Accepted: February 2001?Published online September 3, 2001     

Algebras associated to intermediate subfactors

The Temperley-Lieb algebras are the fundamental symmetry associated to any inclusion of factors with finite index. We analyze in this paper the situation when there is an intermediate subfactor of . The additional symmetry is captured by a tower of certain algebras associated to . These algebras form a Popa system (or standard lattice) and thus, by a theorem of Popa, arise as higher relative commutants of a subfactor. This subfactor gives a free composition (or minimal product) of an and an subfactor. We determine the Bratteli diagram describing their inclusions. This is done by studying a hierarchy of colored generalizations of the Temperley-Lieb algebras, using a diagrammatic approach, à la Kauffman, that is independent of the subfactor context. The Fuss-Catalan numbers appear as the dimensions of our algebras. We give a presentation of the and calculate their structure in the semisimple case employing a diagrammatic method. The principal part of the Bratteli diagram describing the inclusions of the algebras is the Fibonacci graph. Our algebras have a natural trace and we compute the trace weights explicitly as products of Temperley-Lieb traces. If all indices are , we prove that the algebras and coincide. If one of the indices is , is a quotient of and we compute the Bratteli diagram of the tower . Our results generalize to a chain of intermediate subfactors. Oblatum 1-XII-1995 & 1-VII-1996     

Isospectral deformations on Riemannian manifolds which are diffeomorphic to compact Heisenberg manifolds

It is known that ifH _m is the classical (2m+1)-dimensional Heisenberg group, Γ a cocompact discrete subgroup ofH _m andg a left invariant metric, then (Γ/H _{m, g}) is infinitesimally spectrally rigid within the family of left invariant metrics. The purpose of this paper is to show that for everym≥2 and for a certain choice of Γ andg, there is a deformation (Γ/H _{m, g α}) withg=g ₁ such that for every α≠1, (Γ/H _{m, g α})does admit a nontrivial isospectral deformation. For α≠1 the metricsg _α will not beH _m-left invariant, and the (Γ/H _m, gx_α) will not be nilmanifolds, but still solvmanifolds.     

On diagrammatic bounds of knot volumes and spectral invariants

In recent years, several families of hyperbolic knots have been shown to have both volume and λ₁ (first eigenvalue of the Laplacian) bounded in terms of the twist number of a diagram, while other families of knots have volume bounded by a generalized twist number. We show that for general knots, neither the twist number nor the generalized twist number of a diagram can provide two-sided bounds on either the volume or λ₁. We do so by studying the geometry of a family of hyperbolic knots that we call double coil knots, and finding two-sided bounds in terms of the knot diagrams on both the volume and on λ₁. We also extend a result of Lackenby to show that a collection of double coil knot complements forms an expanding family iff their volume is bounded.     

On graded quotient modules of mapping class groups of surfaces

Let Γ _{g, n} be the mapping class group of a compact Riemann surface of genusg withn points preserved (2−2g−n<0,g≥1,n≥0). The Torelli subgroup of Γ _{g, n} has a natural weight filtration {Γ_{g, n}(m)} _m≥1. Each graded quotient gr ^m Γ _{g, n} ⊗ ℚ (m≥1) is a finite dimensional vector space over ℚ on which the group Sp(2g, ℚ)×S _n naturally acts. In this paper, we have determined the Sp(2g, ℚ)×S _n module structure of gr ^m Γ _{g, n} ⊗ ℚ for 1≤m≤3. This includes a verification of an expectation by S. Morita. Also, for generalm, we have identified a certain Sp(2g, ℚ)-irreducible component of gr ^m Γ _{g, n} ⊗ ℚ by constructing explicitly elements in these modules.     

Adic realizations of ergodic actions by homeomorphisms of Markov compacta and ordered Bratteli diagrams

For any ergodic transformation T of a Lebesgue space (X, μ), it is possible to introduce a topology τ on X such that (a) X becomes a totally disconnected compactum (a Cantor set) with a Markov structure, and μ becomes a Borel Markov measure; (b) T becomes a minimal strictly ergodic homeomorphism of (X, τ); (c) the orbit partition of T is the tail partition of the Markov compactum (up to two classes of the partition). The Markov compactum structure is the same as the path structure of the Bratteli diagram for some AF-algebra. Bibliography: 19 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 223, 1995, pp. 120–126.     

TT-eigentensors for the Lichnerowicz Laplacian on some asymptotically hyperbolic manifolds with warped products metrics

Let (M =]0, ∞[×N, g) be an asymptotically hyperbolic manifold of dimension n + 1 ≥ 3, equipped with a warped product metric. We show that there exist no TT L ²-eigentensors with eigenvalue in the essential spectrum of the Lichnerowicz Laplacian Δ _L . If (M, g) is the real hyperbolic space, there is no symmetric L ²-eigentensors of Δ _L .     

The algebra ofG-relations

In this paper, we study a tower {A _n ^G: n} ≥ 1 of finite-dimensional algebras; here, G represents an arbitrary finite group,d denotes a complex parameter, and the algebraA _n ^G(d) has a basis indexed by ‘G-stable equivalence relations’ on a set whereG acts freely and has 2n orbits. We show that the algebraA _n ^G(d) is semi-simple for all but a finite set of values ofd, and determine the representation theory (or, equivalently, the decomposition into simple summands) of this algebra in the ‘generic case’. Finally we determine the Bratteli diagram of the tower {A _n ^G(d): n} ≥ 1 (in the generic case).     

On the hyperbolic limit points of groups acting on hyperbolic spaces

We study the hyperbolic limit points of a groupG acting on a hyperbolic metric space, and consider the question of whether any attractive limit point corresponds to a unique repulsive limit point. In the special case whereG is a (non-elementary) finitely generated hyperbolic group acting on its Cayley graph, the answer is affirmative, and the resulting mapg ⁺↦g ⁻, is discontinuous everywhere on the hyperbolic boundary. We also provide a direct, combinatorial proof in the special case whereG is a (non-abelian) free group of finite type, by characterizing algebraically the hyperbolic ends ofG. Partially supported by a grant from M.U.R.S.T., Italy.     

The Classification of Complex Factor-Representations of the Three-Dimensional Heisenberg Group over a Countable Group Field of Finite Characteristic

We consider a field F that is a direct limit of an increasing chain of finite fields, and describe the Bratteli diagram, complex factor-representations, and projective moduli of the Heisenberg group of 3 × 3 upper-triangular matrices with elements from F. Bibliography: 3 titles.     

Relatively hyperbolic extensions of groups and Cannon-Thurston maps

Let 1 → (K, K ₁) → (G, N _G (K ₁)) → (Q, Q ₁) → 1 be a short exact sequence of pairs of finitely generated groups with K ₁ a proper non-trivial subgroup of K and K strongly hyperbolic relative to K ₁. Assuming that, for all g ∈ G, there exists k _g ∈ K such that gK ₁ g ⁻¹ = k _g K ₁ k _g⁻¹, we will prove that there exists a quasi-isometric section s: Q → G. Further, we will prove that if G is strongly hyperbolic relative to the normalizer subgroup N _G (K ₁) and weakly hyperbolic relative to K ₁, then there exists a Cannon-Thurston map for the inclusion i: Γ _K → Γ _G .     

A New Result on Alspach's Problem

Let G be a simple graph. Let g(x) and f(x) be integer-valued functions defined on V(G) with g(x)≥2 and f(x)≥5 for all x∈V(G). It is proved that if G is an (mg+m−1, mf−m+1)-graph and H is a subgraph of G with m edges, then there exists a (g,f)-factorization of G orthogonal to H. Received: January 19, 1996 Revised: November 11, 1996     

Spectral geometry,link complements and surgery diagrams

We provide an upper bound on the Cheeger constant and first eigenvalue of the Laplacian of a finite-volume hyperbolic 3-manifold M, in terms of data from any surgery diagram for M. This has several consequences. We prove that a family of hyperbolic alternating link complements is expanding if and only if they have bounded volume. We also provide examples of hyperbolic 3-manifolds which require ‘complicated’ surgery diagrams, thereby proving that a recent theorem of Constantino and Thurston is sharp. Along the way, we find a new upper bound on the bridge number of a knot that is not tangle composite, in terms of the twist number of any diagram of the knot. The proofs rely on a theorem of Lipton and Tarjan on planar graphs, and also the relationship between many different notions of width for knots and 3-manifolds.     

The Graver Complexity of Integer Programming

In this article we establish an exponential lower bound on the Graver complexity of integer programs. This provides new type of evidence supporting the presumable intractability of integer programming. Specifically, we show that the Graver complexity of the incidence matrix of the complete bipartite graph K ₃,m satisfies g(m) = Ω(2 ^m ), with g(m) ≥ 17·2 ^m−3 –7 for every m > 3.     

On the global theory of projective mappings

We consider the theory of constant rank projective mappings of compact Riemannian manifolds from the global point of view. We study projective immersions and submersions. As an example of the results, letf:(M, g) → (N, g′) be a projective submersion of anm-dimensional Riemannian manifold (M, g) onto an (m−1)-dimensional Riemannian manifold (N, g′). Then (M, g) is locally the Riemannian product of the sheets of two integrable distributions Kerf _* and (Kerf _*)^⊥ whenever (M, g) is one of the two following types: (a) a complete manifold with Ric ≥ 0; (b) a compact oriented manifold with Ric ≤ 0. Translated fromMatematicheskie Zametki, Vol. 58, No. 1, pp. 111–118, July, 1995. This work was partially supported by the Russian Foundation for Basic Research grant No. 94-01-0195.     

Invariant Radon measures on measured lamination space

Let S be an oriented surface of genus g≥0 with m≥0 punctures and 3g-3+m≥2. We classify all Radon measures on the space of measured geodesic laminations which are invariant under the action of the mapping class group of S.     

Movement of Intransitive Permutation Groups Having Maximum Degree

Let G be a permutation group on a set Ω with no fixed points in,and m be a positive integer.Then the movement of G is defined as move(G):=sup Γ {|Γg\Γ| | g ∈ G}.It was shown by Praeger that if move(G) = m,then |Ω| 3m + t-1,where t is the number of G-orbits on.In this paper,all intransitive permutation groups with degree 3m+t-1 which have maximum bound are classified.Indeed,a positive answer to her question that whether the upper bound |Ω| = 3m + t-1 for |Ω| is sharp for every t > 1 is given.     

Configuration Models For Moduli Spaces of Riemann surfaces with boundary

In this article we consider Riemann surfacesF of genus g ≥ 0 with n ≥ 1 incoming and m ≥ 1 outgoing boundary circles, where on each incoming circle a point is marked. For the moduli space m_g*(m, n) of all suchF of genusg ≥ 0 a configuration space model Rad_h(m, n) is described: it consists of configurations of h = 2g-2+m+n pairs of radial slits distributed over n annuli; certain combinatorial conditions must be satisfied to guarantee the genusg and exactly m outgoing circles. Our main result is a homeomorphism between Rad_h(m, n) and M_g*(m,n). The space Rad_h(m, n) is a non-compact manifold, and the complement of a subcomplex in a finite cell complex. This can be used for homological calculations. Furthermore, the family of spaces Rad_h(m, n ) form an operad, acting on various spaces connected to conformai field theories.     

Topology of spaces of hyperbolic polynomials and combinatorics of resonances

In this paper we study the topology of the strata, indexed by number partitions λ, in the natural stratification of the space of monic hyperbolic polynomials of degreen. We prove stabilization theorems for removing an independent block or an independent relation in λ. We also prove contractibility of the ‘one-point compactifications of the strata indexed by a large class of number partitions, including λ=(k ^m , 1 ^r ), form≥2. Furthermore, we study the maps between the homology groups of the strata, induced by imposing additional relations (resonances) on the number partition λ, or by merging some of the blocks of λ.