Non-amenable finitely presented torsion-by-cyclic groups

Abstract. – We construct a finitely presented non-amenable group without free non-cyclic subgroups thus providing a finitely presented counterexample to von Neumann’s problem. Our group is an extension of a group of finite exponent n ≫ 1 by a cyclic group, so it satisfies the identity [x,y] ⁿ = 1. Manuscrit reĉu le 8 février 2001. RID="*" ID="*"Both authors were supported in part by the NSF grant DMS 0072307. In addition, the research of the first author was supported in part by the Russian Fund for Basic Research 99-01-00894 and by the INTAS grant, the research of the second author was supported in part by the NSF grant DMS 9978802.     

Quasiequational Theories of Flat Algebras

We prove that finite flat digraph algebras and, more generally, finite compatible flat algebras satisfying a certain condition are finitely q-based (possess a finite basis for their quasiequations). We also exhibit an example of a twelve-element compatible flat algebra that is not finitely q-based. The first author was partially supported by the grant # 201/02/0594 of the Grant Agency of the Czech Republic, and by the Institutional grant MSM0021620839; the second author was partially supported by the grant No. Tn37877 of the Hungarian National Foundation for Scientific Research (OTHA); the third author was supported by the NSF grant # DMS-9971352.     

Coloring complexes and arrangements

Steingrimsson’s coloring complex and Jonsson’s unipolar complex are interpreted in terms of hyperplane arrangements. This viewpoint leads to short proofs that all coloring complexes and a large class of unipolar complexes have convex ear decompositions. These convex ear decompositions impose strong new restrictions on the chromatic polynomials of all finite graphs. Similar results are obtained for characteristic polynomials of submatroids of type ℬ _n arrangements. The first author was supported by NSF grant DMS-0500638. The second author was supported by NSF grant DMS-0245623.     

Subgroups of free profinite groups and large subfields of\mathop Q\limits^ \sim

We prove that many subgroups of free profinite groups are free, and use this to give new examples of pseudo-algebraically closed subfields of satisfying Hilbert’s Irreducibility Theorem, and to solve problems posed by M. Jarden and A. Macintyre. We also find a subfield of which does not satisfy Hilbert’s Irreducibility Theorem, but all of whose proper finite extensions do. The first author was supported by NSF grant MCS76-11625.     

A characterization of the complex affine line

Summary A characterization of affine nonsingular complex algebraic curves that are biregularly isomorphic to ℂ is given; it is stated in terms of approximation of holomorphic maps by regular maps. The author was partially supported by an NSF grant and NATO Collaborative Research Grants Programme, CRG 930238. This article was processed by the author using theLatext style file from Springer-Verlag.     

Avoidable structures, I: Finite ordered sets, semilattices and lattices

We find all finite unavoidable ordered sets, finite unavoidable semilattices and finite unavoidable lattices. While working on this paper, the second and third authors were supported by US NSF grant DMS-0604065. The second author was also supported by the Grant Agency of the Czech Republic, grant #201/05/0002 and by the institutional grant MSM0021620839 financed by MSMT.     

A combination theorem for Veech subgroups of the mapping class group

In this paper we prove a combination theorem for Veech subgroups of the mapping class group analogous to the first Klein–Maskit combination theorem for Kleinian groups in which two Fuchsian subgroups are amalgamated along a parabolic subgroup. As a corollary, we construct subgroups of the mapping class group (for all genera at least 2), which are isomorphic to non-abelian closed surface groups in which all but one conjugacy class (up to powers) is pseudo-Anosov. Received: October 2004 Revision: April 2005 Accepted: April 2005 C.J.L.’s work was partially supported by an NSF postdoctoral fellowship. A.W.R’s work was partially supported by an NSF grant.     

Groups with small Dehn functions and bipartite chord diagrams

We introduce a new invariant of bipartite chord diagrams and use it to construct the first examples of groups with Dehn function n²log n. Some of these groups have undecidable conjugacy problem. Our groups are multiple HNN extensions of free groups. We show that n²log n is the smallest Dehn function of a multiple HNN extension of a free group with undecidable conjugacy problem. Both authors were supported in part by the NSF grants DMS 0245600 and DMS 0455881. In addition, the research of the first author was supported in part by the Russian Fund for Basic Research 05-01-00895, the research of the second author was supported in part by the NSF grant DMS 9978802 and the US-Israeli BSF grant 1999298. Received: February 2005; Revision: September 2005; Accepted: September 2005     

Sharp trace theorems for null hypersurfaces on Einstein metrics with finite curvature flux

The main objective of the paper is to prove a geometric version of sharp trace and product estimates on null hypersurfaces with finite curvature flux. These estimates play a crucial role to control the geometry of such null hypersurfaces. The paper is based on an invariant version of the classical Littlewood–Paley theory, in a noncommutative setting, defined via heat flow on surfaces. Received: April 2004 Revision: June 2005 Accepted: July 2005 The first author is partially supported by NSF grant DMS-0070696. The second author is a Clay Prize Fellow and is partially supported by NSF grant DMS-01007791.     

Inverse coefficient problems for nonlinear parabolic differential equations

This paper is devoted to a class of inverse problems for a nonlinear parabolic differential equation. The unknown coefficient of the equation depends on the gradient of the solution and belongs to a set of admissible coefficients. It is proved that the convergence of solutions for the corresponding direct problems continuously depends on the coefficient convergence. Based on this result the existence of a quasisolution of the inverse problem is obtained in the appropriate class of admissible coefficients.     

Interior-point methods for nonconvex nonlinear programming: regularization and warmstarts

In this paper, we investigate the use of an exact primal-dual penalty approach within the framework of an interior-point method for nonconvex nonlinear programming. This approach provides regularization and relaxation, which can aid in solving ill-behaved problems and in warmstarting the algorithm. We present details of our implementation within the loqo algorithm and provide extensive numerical results on the CUTEr test set and on warmstarting in the context of quadratic, nonlinear, mixed integer nonlinear, and goal programming. Research of the first author is sponsored by ONR grant N00014-04-1-0145. Research of the second author is supported by NSF grant DMS-0107450.     

Singularities on complete algebraic varieties

We prove that any finite set of n-dimensional isolated algebraic singularities can be afforded on a simply connected projective variety. The first author was partially supported by NSF grant DMS-01591 and the third author was partially supported by NSF grant DMS-03693     

Wiggly sets and limit sets

We show that a compact, connected set which has uniform oscillations at all points and at all scales has dimension strictly larger than 1. We also show that limit sets of certain Kleinian groups have this property. More generally, we show that ifG is a non-elementary, analytically finite Kleinian group, and its limit set Λ(G) is connected, then Λ(G) is either a circle or has dimension strictly bigger than 1. The first author is partially supported by NSF Grant DMS 95-00577 and an Alfred P. Sloan research fellowship. The second author is partially supported by NSF grant DMS-94-23746.     

Division theorems and twisted complexes

We prove new Skoda-type division, or ideal membership, theorems. We work in a geometric setting of line bundles over Kahler manifolds that are Stein away from an analytic subvariety. (This includes complex projective manifolds.) Our approach is to combine the twisted Bochner-Kodaira Identity, used in the Ohsawa-Takegoshi Theorem, with Skoda’s basic estimate for the division problem. Techniques developed by McNeal and the author are then used to provide many examples of new division theorems. Among other applications, we give a modification of a recent result of Siu regarding effective finite generation of certain section rings. Partially supported by NSF grant DMS-0400909.     

Finite Element Method and Discontinuous Galerkin Method for Stochastic Scattering Problem of Helmholtz Type in ℝ<Superscript><Emphasis Type="Italic">d</Emphasis></Superscript> (<Emphasis Type="Italic">d</Emphasis> = 2, 3)

In this paper, we address the finite element method and discontinuous Galerkin method for the stochastic scattering problem of Helmholtz type in ℝ ^d (d = 2, 3). Convergence analysis and error estimates are presented for the numerical solutions. The effects of the noises on the accuracy of the approximations are illustrated. Results of the numerical experiments are provided to examine our theoretical results. The first author is supported by NSF under grand number 0609918 and AFOSR under grant numbers FA9550-06-1-0234 and FA9550-07-1-0154, the second author is supported by NSFC(10671082, 10626026, 10471054), and the third author is supported by NNSF (No. 10701039 of China) and 985 program of Jilin University.     

An Adaptive Finite Element Method for the H- ψ Formulation of Time-dependent Eddy Current Problems

In this paper, we develop an adaptive finite element method based on reliable and efficient a posteriori error estimates for the H − ψ formulation of eddy current problems with multiply connected conductors. Multiply connected domains are considered by making “cuts”. The competitive performance of the method is demonstrated by an engineering benchmark problem, Team Workshop Problem 7, and a singular problem with analytic solution.W. Zheng was supported in part by China NSF under the grant 10401040.Z. Chen was supported in part by China NSF under the grant 10025102 and 10428105, and by the National Basic Research Project under the grant 2005CB321701.     

The Manin—Drinfeld theorem and Ramanujan sums

The Manin—Drinfeld theorem asserts the finiteness of the cuspidal divisor class group of a modular curve corresponding to a congruence subgroup. The purpose of the note is to draw attention to the connection between this theorem and Ramanujan sums, and to the question of what happens for non-congruence subgroups. Research partially supported by an NSERC grant. Alfred P Sloan Fellow. Research partially supported by the NSF.     

Invariant measures for the horocycle flow on periodic hyperbolic surfaces

We classify the ergodic invariant Radon measures for the horocycle flow on geometrically infinite regular covers of compact hyperbolic surfaces. The method is to establish a bijection between these measures and the positive minimal eigenfunctions of the Laplacian of the surface. Two consequences arise: if the group of deck transformations G is of polynomial growth, then these measures are classified by the homomorphisms from G ₀ to ℝ where G ₀ ≤ G is a nilpotent subgroup of finite index; if the group is of exponential growth, then there may be more than one Radon measure which is invariant under the geodesic flow and the horocycle flow. We also treat regular covers of finite volume surfaces. The first author was supported by NSF grant 0500630. The second author was supported by NSF grant 0400687.     

Weights and covers of chamber systems

The notion of weight algebra for a group geometry is developed for flag-transitive subgroups and their quotients, as a technique for establishing the existence of weight vectors in modular representations. This extends the applicability of earlier results, in particular results for the full group of a spherical building. The method adapts for at least some of the sporadic geometries provided by finite quotients of infinite affine buildings. Dedicated to Professor Jacques Tits on his 60th birthday Partially supported by NSF grant DMS 88-01679 and NSA grant MDA 904-87-H-2006.     

On the number of generators needed for free profinite products of finite groups

We provide lower estimates on the minimal number of generators of the profinite completion of free products of finite groups. In particular, we show that if C ₁, ..., C _n are finite cyclic groups then there exists a finite group G which is generated by isomorphic copies of C ₁, ..., C _n and the minimal number of generators of G is n. The first author’s research is partially supported by NSF grant DMS-0701105. The second author’s research is partially supported by OTKA grant T38059 and the Magyary Zoltán Postdoctoral Fellowship.