The Conjugacy Problem is Solvable in Free-By-Cyclic Groups

We show that the conjugacy problem is solvable in [finitelygenerated free]-by-cyclic groups, by using a result of O. Maslakovathat one can algorithmically find generating sets for the fixedsubgroups of free group automorphisms, and one of P. Brinkmannthat one can determine whether two cyclic words in a free groupare mapped to each other by some power of a given automorphism.We also solve the power conjugacy problem, and give an algorithmto recognize whether two given elements of a finitely generatedfree group are twisted conjugated to each other with respectto a given automorphism. 2000 Mathematics Subject Classification20F10, 20E05.     

Characterizing Automorphism Groups of Ordered Abelian Groups

A proof is given of the following theorem, which characterizesfull automorphism groups of ordered abelian groups: a groupH is the automorphism group of some ordered abelian group ifand only if H is right-orderable. 2000 Mathematics Subject Classification20K15, 20K20, 20F60, 20K30 (primary); 03E05 (secondary).     

Uncountable Homomorphic Images of Polish Groups are not N1-Free Groups

Shelah has recently proved that an uncountable free group cannotbe the automorphism group of a countable structure. In fact,he proved a more general result: an uncountable free group cannotbe a Polish group. A natural question is: can an uncountable₁-free group be a Polish group? A negative answer is given here;indeed, it is proved that an ₁-free group cannot be a homomorphicimage of a Polish group. In fact, a stronger result is proved,involving a non-commutative analogue of the notion of separablegroup. 2000 Mathematics Subject Classification 20E05.     

Group Laws and Free Subgroups in Topological Groups

A proof is given that a permutation group in which differentfinite sets have different stabilizers cannot satisfy any grouplaw. For locally compact topological groups with this property,almost all finite subsets of the group are shown to generatefree subgroups. Consequences of these theorems are derived for:Thompson's group F, weakly branch groups, automorphism groupsof regular trees, and profinite groups with alternating compositionfactors of unbounded degree. 2000 Mathematics Subject Classification20B07, 20E10, 20E18, 20P05.     

Dunkl Operators for Complex Reflection Groups

Dunkl operators for complex reflection groups are defined inthis paper. These commuting operators give rise to a parameterizedfamily of deformations of the polynomial De Rham complex. Thisleads to the study of the polynomial ring as a module over the‘rational Cherednik algebra’, and a natural contravariantform on this module. In the case of the imprimitive complexreflection groups G(m, p, N), the set of singular parametersin the parameterized family of these structures is describedexplicitly, using the theory of non-symmetric Jack polynomials.2000 Mathematical Subject Classification: 20F55 (primary), 52C35,05E05, 33C08 (secondary).     

On Growth of Groups with Few Relators

Let G be a group that has a presentation with n generators andm relators, where m < n, and let S be an arbitrary generatingset for G. Then some subset of n-m elements of S freely generatesa free subgroup of G. This leads to the proof of a conjectureof Gromov on growth of groups. 2000 Mathematics Subject Classification20F05, 20E18.     

Transformations Between Sturm-Liouville Problems with Eigenvalue Dependent and Independent Boundary Conditions

Explicit relationships are given connecting ‘almost’isospectral Sturm-Liouville problems with eigen-value dependent,and independent, boundary conditions, respectively. Applicationis made to various direct and inverse spectral questions. 2000Mathematics Subject Classification 34B24, 34L05, 47E05.     

Exactness of Free and Amenable Groups by the Construction of Ozawa Kernels

Using properties of their Cayley graphs, specific examples ofOzawa kernels are constructed for both free and amenable groups,thus showing that these groups satisfy Property O. It is deducedboth that these groups are exact and satisfy Yu's Property A.2000 Mathematics Subject Classification 20F65 (primary), 46l05(secondary).     

On a Problem of P. Hall about Groups Isomorphic to all their Non-Trivial Normal Subgroups

A scheme of construction of infinite groups, other than simplegroups, free groups of infinite rank and the infinite cyclicgroup, which are isomorphic to all their non-trivial normalsubgroups is presented. Some results about the automorphismgroups of simple infinite groups are also obtained. In particular,it is proved that there is an infinite group G of any sufficientlylarge prime exponent p (or which is torsion-free) all of whoseproper subgroups are cyclic, and such that the groups Aut Gand Out G are isomorphic. The proofs use the technique of gradeddiagrams developed by A. Yu. Ol'shanskii. 1991 Mathematics SubjectClassification: 20F05, 20F06.     

Topology on the Spaces Of Orderings of Groups

A natural topology on the space of left orderings of an arbitrarysemi-group is introduced here. This space is proved to be compact,and for free abelian groups it is shown to be homeomorphic tothe Cantor set. An application of this result is a new proofof the existence of universal Gröbner bases. 2000 MathematicsSubject Classification 06F15, 13P10 (primary), 06F05, 20F60(secondary).     

Modular Subgroup Arithmetic and a Theorem of Philip Hall

A surprising relationship is established in this paper, betweenthe behaviour modulo a prime p of the number S_n G of index nsubgroups in a group G, and that of the corresponding subgroupnumbers for a normal subgroup in G normal subgroup in p-powerorder. The proof relies, among other things, on a twisted versiondue to Philip Hall of Frobenius' theorem concerning the equationx^m=1 in finite groups. One of the applications of this result,presented here, concerns the explicit determination modulo pof S_n G in the case when G is the fundamental group of a treeof groups all of whose vertex groups are cyclic of p-power order.Furthermore, a criterion is established (by a different technique)for the function S_n G to be periodic modulo p. 2000 MathematicsSubject Classification 20E06, 20F99 (primary); 05A15, 05E99(secondary).     

Corrigendum

A UNIQUENESS THEOREM IN THE INVERSE SPECTRAL THEORY OF A CERTAINHIGHER-ORDER ORDINARY DIFFERENTIAL EQUATION USING PALEY–WIENERMETHODS E. ANDERSSON erika{at}maths.lth.se Footnotes 2000 Mathematics Subject Classification 30D15, 34L40, 47E05(primary), 30D10, 34A55, 34B20, 34L05 (secondary). Received September 21, 2006; published online July 8, 2007.     

On the Decidability of Iterated Semidirect Products with Applications to Complexity

The notion of hyperdecidability has been introduced by the firstauthor as a tool to prove decidability of semidirect productsof pseudovarieties of semigroups. In this paper we considersome stronger notions which lead to improved decidability resultsallowing us in turn to establish the decidability of some iteratedsemidirect products. Roughly speaking, the decidability of asemidirect product follows from a mild, commonly verified propertyof the first factor plus the stronger property for all the otherfactors. A key role in this study is played by intermediatefree semigroups (relatively free objects of expanded type lyingbetween relatively free and relatively free profinite objects).As an application of the main results, the decidability of theKrohn–Rhodes (group) complexity is shown to follow fromnon-algorithmic abstract properties likely to be satisfied bythe pseudovariety of all finite aperiodic semigroups, therebysuggesting a new approach to answer (affirmatively) the questionas to whether complexity is decidable. 1991 Mathematics SubjectClassification: primary 20M05, 20M07, 20M35; secondary 08B20.     

The Gallery Length Filling Function and a Geometric Inequality for Filling Length

We exploit duality considerations in the study of singular combinatorial2-discs (diagrams) and are led to the following innovationsconcerning the geometry of the word problem for finite presentationsof groups. We define a filling function called gallery lengththat measures the diameter of the 1-skeleton of the dual ofdiagrams; we show it to be a group invariant and we give upperbounds on the gallery length of combable groups. We use gallerylength to give a new proof of the Double Exponential Theorem.Also we give geometric inequalities relating gallery lengthto the space-complexity filling function known as filling length.2000 Mathematics Subject Classification 20F05 (primary), 20F06,57M05, 57M20 (secondary).     

A New Look at the Burnside-Schur Theorem

The famous Burnside–Schur theorem states that every primitivefinite permutation group containing a regular cyclic subgroupis either 2-transitive or isomorphic to a subgroup of a 1-dimensionalaffine group of prime degree. It is known that this theoremcan be expressed as a statement on Schur rings over a finitecyclic group. Generalizing the latter, Schur rings are introducedover a finite commutative ring, and an analogue of this statementis proved for them. Also, the finite local commutative ringsare characterized in permutation group terms. 2000 MathematicsSubject Classification 20B10, 20B15, 05E99.     

On Subgroups of Finite Index in Positively Finitely Generated Groups

This paper proves that a subgroup of finite index in a positivelyfinitely generated profinite group has maximal subgroup growthat most n^log(n). In particular such a subgroup cannot be free,answering a question by L. Pyber. 2000 Mathematics Subject Classification20E28, 20P05.     

Sturmian morphisms,the braid group <Emphasis Type="Italic">B</Emphasis><Subscript>4</Subscript>, Christoffel words and bases of <Emphasis Type="Italic">F</Emphasis><Subscript>2</Subscript>

We give a presentation by generators and relations of a certain monoid generating a subgroup of index two in the group Aut(F ₂) of automorphisms of the rank two free group F ₂ and show that it can be realized as a monoid in the group B ₄ of braids on four strings. In the second part we use Christoffel words to construct an explicit basis of F ₂ lifting any given basis of the free abelian group Z ². We further give an algorithm allowing to decide whether two elements of F ₂ form a basis or not. We also show that, under suitable conditions, a basis has a unique conjugate consisting of two palindromes. Mathematics Subject Classification (2000) 05E99, 20E05, 20F28, 20F36, 20M05, 37B10, 68R15     

Existence of Solutions for Asymptotically 'Linear' p-Laplacian Equations

Under very general conditions, a proof is given, via Morse theory,of the existence of solutions for asymptotically ‘linear’p-Laplacian equations, where the asymptotic limit may be greaterthan the second eigenvalue. The existence of nonzero solutionsis also considered. 2000 Mathematics Subject Classification35J65, 58E05.     

Hecke Algebras and SLn-Types

Let F be a non-Archimedean local field. In an earlier paper,we constructed certain types (in the sense of Bushnell and Kutzko)in SL_n (F), in fact enough to describe the non-supercuspidalcomponents of the Bernstein decomposition of this group. Wenow give an almost fully explicit presentation of the Heckealgebras of these types. 2000 Mathematics Subject Classification22E50, 20G05.     

Latin Squares and the Hall-Paige Conjecture

The Hall–Paige conjecture deals with conditions underwhich a finite group G will possess a complete mapping, or equivalentlya Latin square based on the Cayley table of G will possess atransversal. Two necessary conditions are known to be: (i) thatthe Sylow 2-subgroups of G are trivial or non-cyclic, and (ii)that there is some ordering of the elements of G which yieldsa trivial product. These two conditions are known to be equivalent,but the first direct, elementary proof that (i) implies (ii)is given here. It is also shown that the Hall–Paige conjecture impliesthe existence of a duplex in every group table, thereby provinga special case of Rodney's conjecture that every Latin squarecontains a duplex. A duplex is a ‘double transversal’,that is, a set of 2n entries in a Latin square of order n suchthat each row, column and symbol is represented exactly twice.2000 Mathematics Subject Classification 05B15, 20D60.