Compact deformations of Fuchsian groups

A conformal map Φ on the unit disk is called a compact deformation of a Fuchsian groupG if Φ has a quasiconformal extension to the planeh which conjugatesG to a Kleinian group G′ and the dilatation ofh is compactly supported moduloG. We show that for such deformations δ = dim(∧(G′)) = dim(∧_c(G′)) (if δ ≥1) and the image of ∧_e = ∧ ∧_c is contained in a countable union of rectifiable curves and has zero length iffG is divergence type. The first author is partially supported by NSF Grant DMS 01-03626. The second author is partially supported by NSF grant DMS-99-71311.     

Primes dividing the degrees of the real characters

Let G be a finite group and let Irr(G) denote the set of all complex irreducible characters of G. The Ito–Michler Theorem asserts that if a prime p does not divide the degree of any χ Irr(G) then a Sylow p-subgroup P of G is normal in G. We prove a real-valued version of this theorem, where instead of Irr(G) we only consider the subset Irr_rv(G) consisting of all real-valued irreducible characters of G. We also prove that the character degree graph associated to Irr_rv(G) has at most 3 connected components. Similar results for the set of real conjugacy classes of G have also been obtained. Part of this paper was done while the second author visited the Mathematics Department of the Università di Firenze. He would like to thank the Department for its hospitality. The authors are also grateful to F. Lübeck for helping them with some computer calculations. The research of the first author was partially supported by MIUR research program “Teoria dei gruppi ed applicazioni”. This research of the second author was partially supported by the Spanish Ministerio de Educación y Ciencia proyecto MTM2004-06067-C02-01. The third author gratefully acknowledges the support of the NSA and the NSF.     

A Kleinian group with contractible quotient not simply connected at infinity

We give an example of a co-compact Kleinian group Γ which contains a subgroupΓ ₀ having the property that ℝ³/Γ ₀ is contractible but not simply connected at infinity. Research partially supported by NSF. Research partially supported by NSF and Sloan Foundation.     

Combinatorial dimension and random sets

Having defined the combinatorial dimension of an arbitrary subset of a finite dimensional lattice, for every αε(1,2) we produce a set in N² whose dimension equals α. Research partially supported by NSF Grant #MCS8002716. Research supported by a University of Connecticut Research Grant (Sept. 1980).     

On Connected [g,f +1]-Factors in Graphs

Let G = (V (G),E(G)) be a graph with vertex set V (G) and edge set E(G), and g and f two positive integral functions from V (G) to Z⁺-{1} such that g(v) ≤ f(v) ≤ d_G(v) for all v ∈V (G), where d_G(v) is the degree of the vertex v. It is shown that every graph G, including both a [g,f]-factor and a hamiltonian path, contains a connected [g,f +1]-factor. This result also extends Kano’s conjecture concerning the existence of connected [k,k+1]-factors in graphs. * The work of this author was supported by NSFC of China under Grant No. 10271065, No. 60373025. † The work of these authors was also supported in part by the US Department of Energy’s Genomes to Life program (http://doegenomestolife.org/) under project, “Carbon Sequestration in Synechococcus sp.: From Molecular Machines to Hierarchical Modeling” (www.genomes2life.org) and by National Science Foundation (NSF/DBI-0354771,NSF/ITR-IIS-0407204).     

Commuting elements and spaces of homomorphisms

This article records basic topological, as well as homological properties of the space of homomorphisms Hom(π,G) where π is a finitely generated discrete group, and G is a Lie group, possibly non-compact. If π is a free abelian group of rank equal to n, then Hom(π, G) is the space of ordered n–tuples of commuting elements in G. If G = SU(2), a complete calculation of the cohomology of these spaces is given for n = 2, 3. An explicit stable splitting of these spaces is also obtained, as a special case of a more general splitting. Alejandro Adem was partially supported by the NSF and NSERC. Frederick R. Cohen was partially supported by the NSF, grant number 0340575.     

Automatic continuity of homomorphisms and fixed points on metric compacta

Let G₂(ℝ) × Sp₆(ℝ) and G₂(ℝ) × F₄(ℝ) be split dual pairs in split E₇(ℝ) and E₈(ℝ), respectively. It is known that the exceptional correspondences for these dual pairs are functorial on the level of infinitesimal characters. In this paper we show that these dual pair correspondences are functorial for the minimal K-types of principal series representations. Gordan Savin’s research is partially supported be NSF Grant no. DMS-0551846     

Harmonicity of Quasiconformal Measures and Poisson Boundaries of Hyperbolic Spaces

We consider a group Γ of isometries acting on a proper (not necessarily geodesic) δ -hyperbolic space X. For any continuous α-quasiconformal measure ν on ∂X assigning full measure to Λ _r , the radial limit set of Γ, we produce a (nontrivial) measure μ on Γ for which ν is stationary. This means that the limit set together with ν forms a μ-boundary and ν is harmonic with respect to the random walk induced by μ. As a basic example, take and Γ to be any geometrically finite Kleinian group with ν a Patterson-Sullivan measure for Γ. In the case when X is a CAT(−1) space and Γ is discrete with quasiconvex action, we show that (Λ _r , ν) is the Poisson boundary for μ. In the course of the proofs, we establish sufficient conditions for a set of continuous functions to form a positive basis, either in the L ¹ or L ^∞ norm, for the space of uniformly positive lower-semicontinuous functions on a general metric measure space. The first author was supported in part by an NSF postdoctoral fellowship and DMS-0420432. The second author was supported in part by an NSF postdoctoral fellowship.     

κ-Gorenstein Modules

Let A and F be artin algebras and ∧UГa paper, we first introduce the notion of k-Gorenstein faithfully balanced selforthogonal bimodule. In this modules with respect to ∧UГ and then characterize it in terms of the U-resolution dimension of some special injective modules and the property of the functors Ext^i （Ext^i （-, U）, U） preserving monomorphisms, which develops a classical result of Auslander. As an application, we study the properties of dual modules relative to Gorenstein bimodules. In addition, we give some properties of ∧UГwith finite left or right injective dimension.     

The diameter of paired-domination vertex critical graphs

In this paper we continue the study of paired-domination in graphs introduced by Haynes and Slater (Networks 32 (1998), 199–206). A paired-dominating set of a graph G with no isolated vertex is a dominating set of vertices whose induced subgraph has a perfect matching. The paired-domination number of G, denoted by γ _pr(G), is the minimum cardinality of a paired-dominating set of G. The graph G is paired-domination vertex critical if for every vertex v of G that is not adjacent to a vertex of degree one, γ _pr(G − v) < γ _pr(G). We characterize the connected graphs with minimum degree one that are paired-domination vertex critical and we obtain sharp bounds on their maximum diameter. We provide an example which shows that the maximum diameter of a paired-domination vertex critical graph is at least 3/2 (γ _pr(G) − 2). For γ _pr(G) ⩽ 8, we show that this lower bound is precisely the maximum diameter of a paired-domination vertex critical graph. The first author was supported in part by the South African National Research Foundation and the University of KwaZulu-Natal, the second author was supported by the Natural Sciences and Engineering Research Council of Canada.     

A note on stationarity of spherical measures

In this short note we show that any smooth probability measure on the boundaryB(G) of a semisimple Lie groupG is stationary for some probability measure on a lattice Γ. This generalizes a result of Furstenberg about Poisson boundaries for semisimple Lie groups. The author is supported in part by NSF postdoctoral fellowship.     

Pluricanonical systems on irregular 3-folds of general type

In this paper we prove that if X is an irregular 3-fold with χ(ω _X ) > 0, then |mK _X | is birational for all m ≥ 5.The first author was partially supported by NSC and NCTS of Taiwan. The second author was partially supported by NSF research grant no: 0456363.     

Local sidon sets and uniform convergence of fourier series

For compact groups several necessary and sufficient conditions for a set to be local Sidon are given; these conditions are expressed in functional-analytic terms. An immediate corollary is the existence of a large class of compact groupsG, including all the connected non-Abelian Lie groups, which support functions not inA(G) but having Fourier series that are both unformly and absolutely convergent. This has been shown for the special case ofSU(2) by Mayer. Also several necessary and sufficient conditions for a set to be local Λ (p) are given.     

On a graph packing conjecture by Bollobás,Eldridge and Catlin

Two graphs G ₁ and G ₂ of order n pack if there exist injective mappings of their vertex sets into [n], such that the images of the edge sets are disjoint. In 1978, Bollobás and Eldridge, and independently Catlin, conjectured that if (Δ(G ₁) + 1)(Δ(G ₂) + 1) ≤ n + 1, then G ₁ and G ₂ pack. Towards this conjecture, we show that for Δ(G ₁),Δ(G ₂) ≥ 300, if (Δ(G ₁) + 1)(Δ(G ₂) + 1) ≤ 0.6n + 1, then G ₁ and G ₂ pack. This is also an improvement, for large maximum degrees, over the classical result by Sauer and Spencer that G ₁ and G ₂ pack if Δ(G ₁)Δ(G ₂) < 0.5n. This work was supported in part by NSF grant DMS-0400498. The work of the second author was also partly supported by NSF grant DMS-0650784 and grant 05-01-00816 of the Russian Foundation for Basic Research. The work of the third author was supported in part by NSF grant DMS-0652306.     

Group rings of hyperabelian groups

Ifk is a field of characteristicp>0 then we find all hyper-(Abelian or locally finite-p′) groupsG such that the augmentation ideal of the group algebrakG has the AR property. The second author is partially supported by NSF grant MCS-7828082.     

The space of surface group representations

In this note we prove that the number of irreducible components of Hom (π,G) is the same as π₁(G), where π is a surface group andG is complex semisimple. This is established by studying the flat bundles on Riemann surfaces. The present work is partially supported by NSF grant DMS89-04922     

Noded Teichmüller spaces

LetG be a finitely generated Kleinian group and let Δ be an invariant collection of components in its region of discontinuity. The Teichmüller spaceT(Δ,G) supported in Δ is the space of equivalence classes of quasiconformal homeomorphisms with complex dilatation invariant underG and supported in Δ. In this paper we propose a partial closure ofT(Δ,G) by considering certain deformations of the above hemeomorphisms. Such a partial closure is denoted byNT(Δ,G) and called thenoded Teichmüller space ofG supported in Δ. Some concrete examples are discussed. Partially supported by Projects Fondecyt 1030252, 1030373, 1040333, Projects UTFSM 12.05.21, 12.05.23 and by grant of the University of Bergen.     

Sum-free sets in abelian groups

We show that there is an absolute constant δ>0 such that the number of sum-free subsets of any finite abelian groupG is

On finite Moufang polygons

We show that a finite generalized polygon Γ is Moufang with respect to a groupG if and only if for every flag {x, y} of Γ, the subgroupG ₁(x, y) ofG fixing every element incident with one ofx, y acts transitively on the set of apartments containing the elementsu, x, y, w, whereu≠y (resp.w≠x) is an arbitrary element incident withx (resp.y). Research Associate at the National Fund of Scientific Research of Belgium. Research partially supported by NSF Grant DMS-8901904.