Sol Geometry Groups are not Asynchronously Automatic

Lattices in the 3-dimensional Lie group Sol are shown to benot asynchronously automatic. The proof consists of a combinationof geometry and logical complexity arguments. 2000 MathematicsSubject Classification: 20F10, 20F16.     

Infinite Families of Regular, Aspherical Two-Complexes with the Same Euler Characteristic

A regular 2-complex has a group of cellular automorphisms whichis transitive on 2-cells. Let p bean odd prime. By generalizingan example of Floyd and Parry [5], we obtain infinite familiesof compact, aspherical, regular 2-complexes with Euler characteristicp and infinite hyperbolic fundamental group. 1991 MathematicsSubject Classification 20F32, 57M20, 05C25.     

Peripheral fillings of relatively hyperbolic groups

In this paper a group theoretic version of Dehn surgery is studied. Starting with an arbitrary relatively hyperbolic group G we define a peripheral filling procedure, which produces quotients of G by imitating the effect of the Dehn filling of a complete finite volume hyperbolic 3-manifold M on the fundamental group π₁(M). The main result of the paper is an algebraic counterpart of Thurston’s hyperbolic Dehn surgery theorem. We also show that peripheral subgroups of G ‘almost’ have the Congruence Extension Property and the group G is approximated (in an algebraic sense) by its quotients obtained by peripheral fillings. Mathematics Subject Classification (2000) 20F65, 20F67, 20F06, 57M27, 20E26     

Engel-4 Groups of Exponent 5

We show that if G is a group of exponent 5, and if G satisfiesthe Engel-4 identity [x,y,y,y,y]=1, then G is locally finite.By a result of Traustason, this implies that Engel-4 5-groupsare locally finite. We also show that a group of exponent 5is locally finite if and only if it satsifies the identity This result implies that a group of exponent 5 is locally finiteif its three generator subgroups are finite. 1991 MathematicsSubject Classification: 20D15, 20F45, 20F50.     

Bounds on Volume Increase Under Dehn Drilling Operations

In this paper we investigate how the volume of a hyperbolicmanifold increases under the process of removing a curve, thatis, Dehn drilling. If the curve we remove is a geodesic, weshow that for a certain family of manifolds the volume increaseis bounded above by · l where l is the length of thegeodesic drilled. We further construct examples to show thatthere is no lower bound to the volume increase in terms of amonotonic unbounded function of length. In particular, thisshows that volume increase is not bounded below linearly inlength. 1991 Mathematics Subject Classification: primary 51M10,51M20; secondary 51M25, 52C25.     

The Dehn Function of Stallings’ Group

We prove that the Dehn function of a group of Stallings that is finitely presented but not of type F₃{\mathcal{F}_3} is quadratic.     

An isoperimetric function for Bestvina-Brady groups

Given a right-angled Artin group A, the associated Bestvina–Bradygroup is defined to be the kernel of the homomorphism A thatmaps each generator in the standard presentation of A to a fixedgenerator of . We prove that the Dehn function of an arbitraryfinitely presented Bestvina–Brady group is bounded aboveby n⁴. This is the best possible universal upper bound.     

On 4-manifolds with zero-entropy metrics

We sharpen a recent result of Paternain and Petean by showingthat a closed 4-manifold which admits a Riemannian metric withzero topological entropy and infinite fundamental group eitheris aspherical or has a finite covering space homeomorphic toS³ x S¹ or S² x S¹ x S¹.     

Embeddings of Sz(32) in E8(5)

We show that the Suzuki group Sz(32) is a subgroup of E₈(5),and so is its automorphism group. Both are unique up to conjugacyin E₈(F) for any field F of characteristic 5, and the automorphismgroup Sz(32):5 is maximal in E₈(5). 1991 Mathematics SubjectClassification 20E28.     

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).     

Representing Dehn Twists with Branched Coverings

We show that any homologically nontrivial Dehn twist of a compactsurface F with boundary is the lifting of a half-twist in thebraid group , with respectto a suitable branched covering p : F B². In particular, weallow the surface to have disconnected boundary. As a consequence,any allowable Lefschetz fibration on B² is a branched coveringof B² x B².     

Linear Bounds for the Degree of Subgroup Growth in Terms of the Hirsch Length

Let G be a residually-finite virtually soluble minimax group.We give upper and lower linear bounds for the degree of subgroupgrowth of G in terms of the Hirsch length of G. 2000 MathematicsSubject Classification 20F99.     

Locally Finite Finitary Skew Linear Groups

Let V be a vector space over the division ring D of infinitedimension. We study locally finite, primitive groups G of finitarylinear automorphisms of V. We show that the derived group G'of G is infinite, simple, and lies in every non-trivial normalsubgroup of G, and that G' G Aut G'. Moreover if char D =0, then G is either the finitary symmetric group or the alternatinggroup on some infinite set. If D is commutative, that is, ifD is a field, then all these results are known and are the consequenceof the collective work of a number of people: in particular(in alphabetical order) V. V. Belyaev, J. I. Hall, F. Leinen,U. Meierfrankenfeld, R. E. Phillips, O. Puglisi, A. Radfordand quite probably others. 2000 Mathematics Subject Classification:20H25, 20H20, 20F50.     

A Calculation of the Culler-shalen Seminorms Associated to Small Seifert Dehn Fillings

We determine the relationship between the multiplicities ofthe zeros of certain rational functions defined on the SL₂(C)and PSL₂C character varieties of knot exteriors. This leadsto a formula for the Culler–Shalen seminorms associatedto small Seifert Dehn fillings of such manifolds. 2000 MathematicsSubject Classification: 57M05, 57M27, 57R65.     

Generating the surface mapping class group by two elements

Wajnryb proved in 1996 that the mapping class group of an orientable surface is generated by two elements. We prove that one of these generators can be taken as a Dehn twist. We also prove that the extended mapping class group is generated by two elements, again one of which is a Dehn twist. Another result we prove is that the mapping class groups are also generated by two elements of finite order.

     

The Mathieu Group M11 and the Modular Curve X(11)

In this paper, we prove that the modular curve X(11) over afield of characteristic 3 admits the Mathieu group M₁₁ as anautomorphism group. We also examine some aspects of the geometryof the curve X(11) in characteristic 3. In particular, we showthat every point of the curve is a point of inflection, thecurve has 110 hyperflexes and there are no inflectional trianglesand 11232 inflectional pentagons, of which 144 are self-conjugate.The hyperflexes correspond to the supersingular elliptic curves.We comment on the relationship of Ward's quadrilinear invariantfor M₁₂ to our work and announce for the first time the equationsfor Klein's A-curve of level 11. We also comment on the relationof our work to some unpublished work of Bott and Tate. 1991Mathematics Subject Classification: 11F32, 11G20, 14G10, 14H10,14N10, 20B25, 20C34.     

A Banach Space Whose Operator Algebra has K0-Group Q

We construct a Banach space X whose algebra of bounded linearoperators has K0-group the additive group of rational numbers.We achieve this using a Gowers-Maurey space. 2000 Mathematical Subject Classification: 20F36, 55P35, 55Q05,55Q40, 55U10     

On the Asphericity of Length Five Relative Group Presentations

We investigate asphericity of the relative group presentation G,t |atbtctdtet=1 and prove it aspherical provided thesubgroupof G generated by ab^–1, bc^–1, cd^–1, de^–1is neither finite cyclic nor a finite triangle group. We alsoprove a similar result for the closely related relative grouppresentation G,s,t | sßst=1=tts^–1. 2000 MathematicsSubject Classification: 20F05, 57M05.     

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).     

Noetherian Semigroup Algebras

A complete structural characterization of submonoids S of apolycyclic-by-finite group such that the semigroup algebra K[S]over a field K is right noetherian is obtained. It follows thatsuch algebras are also left noetherian. 2000 Mathematics SubjectClassification 16P40, 16S36, 20M25 (primary), 20F22, 20C07,20M10 (secondary).