the group of isometries of a banach space and duality we construct an example of a real banach space whose group of surjective isometries has no uniformly continuous one parameter semigroups but the group of surjective isometries of its dual contains infinitely many of them other examples concerning numerical index hermitian operators and dissipative operators are also shown

on shift dynamics for cyclically presented groups abstract a group defined by a finite presentation with cyclic symmetry admits a shift automorphism that is periodic and word length preserving it is shown that if the presentation is combinatorially aspherical and orientable in the sense that no relator is a cyclic permutation of the inverse of any of its shifts then the shift acts freely on the non identity elements of the group presented for cyclic presentations defined by positive words of length at most three the shift defines a free action if and only if the presentation is combinatorially aspherical and the shift itself is fixed point free if and only if the group presented is infinite

normal convexity and equations over groups a subgroups of a grouph is said to be normal convex inh if for any subsetr s the natural maps r s h r his injective

algorithmically insoluble problems about finitely presented solvable groups lie and associative algebras 1 1 in 1955 novikov 111 constructed a finitely presented group with an insoluble word problem cf miller lo and also boone 8 shortly afterwards first adyan l 4 and then rabin 13 143 used this group of novikov to prove that for every one of a host of group theoretic properties there is no algorithm whereby one can decide whether any finitely presented group has such a property in particular there is no algorithm whereby one can decide whether any finitely presented group is of order 1 thus the isomorphism problem for finitely presented groups is algorithmically insoluble these negative results were followed in 1959 by a number of similar ones concerned with elements and subgroups of finitely presented groups baumslag boone and neumann s thus for example there is no algorithm whereby one can decide whether or not any element in a suitably chosen finitely presented group is a proper power in 1981 harlampovich 9 constructed a finitely presented solvable group of derived length 3 with an insoluble word problem this then raised the possibility that many of the results cited above for finitely presented groups can be carried over to finitely presented solvable groups it is not possible to carry over these results en masse to finitely presented solvable groups since it is easy to decide whether any finitely presented solvable group is of order 1 indeed there is an algorithm whereby one can determine whether any finitely presented solvable group is polycyclic baumslag cannonito and miller 6 again no such algorithm exists for finitely presented groups as a whole adyan 2 the methods employed previously cannot be applied to solvable groups because the constructions

non orientable surface plus one relation groups recently dicks linnell determined the l 2 betti numbers of the orientable surfaceplus one relation groups and their arguments involved some results that were obtained topologically by hempel and howie using algebraic arguments we now extend all these results of hempel and howie to a larger class of two relator groups and we then apply the extended results to determine the l 2 betti numbers of the non orientable surface plus one relation groups 2000 mathematics subject classification primary 20f05 secondary 16s34 20j05

approximations of groups characterizations of sofic groups and equations over groups we give new characterizations of sofic groups a group g is sofic if and only if it is a subgroup of a quotient of a direct product of alternating or symmetric groups a group g is sofic if and only if any system of equations solvable in all alternating groups is solvable over g the last characterization allows to express soficity of an existentially closed group by sentences

algorithmic problems for free abelian times free groups we study direct products of free abelian and free groups with special emphasis on algorithmic problems after giving natural extensions of standard notions into that family we find an explicit expression for an arbitrary endomorphism of zm fn these tools are used to solve several algorithmic problems for zm fn the membership problem the isomorphism problem the finite index problem the subgroup and coset intersection problems the fixed point problem and the whitehead problem

on the polynomial numerical index of the real spaces c0 l1 and l abstract we give a lower bound for the polynomial numerical index of order k for real lush spaces we use this bound to compute the polynomial numerical index of order 2 of the real spaces c 0 l 1 and l finally we present an example of a real banach space x whose polynomial numerical indices are positive while the ones of its bidual are zero

on asymptotically hereditarily aspherical groups we undertake a systematic study of asymptotically hereditarily aspherical aha groups the class of groups introduced by tadeusz januszkiewicz and the second author as a tool for exhibiting exotic properties of systolic groups we provide many new examples of aha groups also in high dimensions we relate the aha property with the topology at infinity of a group and deduce in this way some new properties of weakly systolic groups we also exhibit an interesting property of boundaries at infinity for a few classes of aha groups

coherence subgroup separability and metacyclic structures for a class of cyclically presented groups abstract we study a class m of cyclically presented groups that includes both finite and infinite groups and is defined by a certain combinatorial condition on the defining relations this class includes many finite metacyclic generalized fibonacci groups that have been previously identified in the literature by analyzing their shift extensions we show that the groups in the class m are coherent subgroup separable satisfy the tits alternative possess finite index subgroups of geometric dimension at most two and that their finite subgroups are all metacyclic many of the groups in m are virtually free some are free products of metacyclic groups and free groups and some have geometric dimension two we classify the finite groups that occur in m giving extensive details about the metacyclic structures that occur and we use this to prove an earlier conjecture concerning cyclically presented groups in which the relators are positive words of length three we show that any finite group in the class m that has fixed point free shift automorphism must be cyclic

injective labeled oriented trees are aspherical a labeled oriented tree is called injective if each vertex occurs at most once as an edge label we show that injective labeled oriented trees are aspherical the proof uses a new relative asphericity test based on a lemma of stallings

efficient finite groups arising in the study of relative asphericity we study a class of two generator two relator groups denoted j n m k that arise in the study of relative asphericity as groups satisfying a transitional curvature condition particular instances of these groups occur in the literature as finite groups of intriguing orders here we find infinite families of non elementary virtually free groups and of finite metabelian non nilpotent groups for which we determine the orders all mersenne primes arise as factors of the orders of the non metacyclic groups in the class as do all primes from other conjecturally infinite families of primes we classify the finite groups up to isomorphism and show that our class overlaps and extends a class of groups f a b c with trivalent cayley graphs that was introduced by c m campbell h s m coxeter and e f robertson the theory of cyclically presented groups informs our methods and we extend part of this theory namely on connections with polynomial resultants to bicyclically presented groups that arise naturally in our analysis as a corollary to our main results we obtain new infinite families of finite metacyclic generalized fibonacci groups

on balanced presentations of the trivial group to solve problems of magnus and stallings stated in the 1960s we construct a balanced presentation of the trivial group such that no defining relator of this presentation could be replaced by a free generator and the group given by the altered presentation would still be trivial some other related problems on balanced presentations of the trivial group are discussed

the group g t e when g is torsion free abstract in this paper we discuss classes of nontrivial elements in the group with a relative presentation g t e where g is torsion free and e is solvable over g

automorphisms of free groups have finitely generated fixed point sets an automorphism of a finitely generated free group f extends to a homeomorphism of the end completion e of f the set of fixed points of this homeomorphism is finitely generated in a certain sense in particular this implies that the subgroup of elements fixed by the automorphism is finitely generated in the usual sense the emphasis on e instead of f in this paper is analogous to thurston s study of surface groups where measured laminations are studied instead of simple closed curves the techniques in this paper arose from a study of the behaviour of an automorphism under iteration that is the dynamics of the automorphism see 11 i thank bill thurston for explaining some of his ideas to me for analysing automorphisms of free groups in particular the result on bounded cancellation below is due to thurston and matt grayson peter scott asked whether the fixed subgroup is always finitely generated in 3 and gersten first answered this question in 2 using quite different methods

lushness numerical index one and duality we construct an example of a banach space which is not lush but whose dual space is lush this example shows that lushness is not equivalent to numerical index one a characterization of lushness for some quotient spaces of l1 spaces and new results on c rich subspaces of scalar or vector valued c k spaces are also presented

stable representatives for symmetric automorphisms of groups and the general form of the scott conjecture let g be a group acting on a tree x such that all edge stabilizers are finite we extend bestvina handel s theory of train tracks for automorphisms of free groups to automorphisms of g which permute vertex stabilizers using this extension we show that there is an upper bound depending only on g for the complexity of the graph of groups decomposition of the fixed subgroups of such automorphisms of g

slicely countably determined banach spaces we introduce the class of slicely countably determined banach spaces which contains in particular all spaces with the radon nikod m property and all spaces without copies of l 1 we present many examples and several properties of this class we give some applications to banach spaces with the daugavet and the alternative daugavet properties lush spaces and banach spaces with numerical index 1 in particular we show that the dual of a real infinite dimensional banach space with the alternative daugavet property contains l 1 and that operators which do not fix copies of l 1 on a space with the alternative daugavet property satisfy the alternative daugavet equation

polynomial numerical indices of banach spaces with 1 unconditional bases abstract the only infinite dimensional complex space with 1 unconditional basis which has polynomial numerical index of order 2 equal to 1 is c0 in the real case there is no space of this type we also show that in the complex case if x is an infinite dimensional banach sequence space with absolute norm whose kothe dual is norming and has polynomial numerical index of order 2 equal to 1 then c 0 x l in the real case again there is no space of this type

a new test for asphericity and diagrammatic reducibility of group presentations we present a new test for studying asphericity and diagrammatic reducibility of group presentations our test can be applied to prove diagrammatic reducibility in cases where the classical weight test fails we use this criterion to generalize results of j howie and s m gersten on asphericity of lots and of adian presentations and derive new results on solvability of equations over groups we also use our methods to investigate a conjecture of s v ivanov related to kaplansky s problem on zero divisors we strengthen ivanov s result for locally indicable groups and prove a weak version of the conjecture

every group is the outer automorphism group of an hnn extension of a fixed triangle group abstract fix an equilateral triangle group t i a b a i b i a b i with i 6 arbitrary our main result is for every presentation p of every countable group q there exists an hnn extension t p of t i such that out t p q we construct the hnn extensions explicitly and examples are given the class of groups constructed have nice categorical and residual properties in order to prove our main result we give a method for recognising malnormal subgroups of small cancellation groups and we introduce the concept of malcharacteristic subgroups

topology of finite graphs this paper derives from a course in group theory which i gave at berkeley in 1982 i wanted to prove the standard theorems on free groups and discovered that after a few preliminaries the notion of locally injective map or immersion of graphs was very useful this enables one to see in an effective easy algorithmic way just what happens with finitely generated free groups one can understand in this way 1 howson s theorem that if a and b are finitely generated subgroups of a free group then a b is finitely generated and 2 m hall s theorem that finitely generated subgroups of free groups are closed in the profinite topology during this course s m gersten came up with a simple proof of h neumann s inequality on the ranks in howson s theorem one of the ideas in gersten s proof was to use core graphs graphs with no trees hanging on subsequently i found that some consequences of a paper of greenberg s could be proved using core graphs and covering translations of immersions the most striking such result is that if a and b are finitely generated subgroups of a free group and if a c b is of finite index in both a and b then a b is of finite index in a v b the subgroup generated by a ub

plane curves associated to character varieties of 3 manifolds consider a compact 3 manifold m with boundary consisting of a single torus the papers cs1 cs2 cgls discuss the variety of characters of sl2 c representations of zl m and some of the ways in which the topological structure of m is reflected in the algebraic geometry of the character variety we will describe in this paper a certain affine algebraic curve dm which is naturally associated to the character variety of m a basis b 9 1 for the peripheral subgroup of m determines an embedding p8 of dm into c x c with coordinates l and m the closure in c 2 of ps dm is a plane algebraic curve and therefore is defined by a polynomial am n0 m that after certain normalizations is uniquely determined up to multiplication by constants the polynomial am b is effectively computable and is an invariant of the manifold m together with the choice of basis b the results in this paper describe how geometric properties of the character variety and hence topological properties of m are reflected by the polynomial am b in the case that m is the complement of a knot k in a homology 3 sphere we may take the basis b to consist of the longitude and meridian of k with the usual orientation conventions this basis is well defined modulo the involution which inverts both the longitude and meridian it will follow from the construction of dm that if the basis b is obtained from b by inverting both generators then the regular maps pb and pb have the same image thus ar am s am b is an invariant of the knot the polynomial am b displays in a striking way information about the incompressible surfaces in m this involves the newton polygon of am b

fixed points of automorphisms of free groups abstract the set of fixed points of an automorphism of a finitely generated free group is a finitely generated group settling a conjecture of g p scott s

some geometric properties of disk algebras abstract in this paper we study some geometrical properties of certain classes of uniform algebras in particular the ball algebra a u b x of all uniformly continuous functions on the closed unit ball and holomorphic on the open unit ball of a complex banach space x we prove that a u b x has k numerical index 1 for every k the lushness and also the ahsp moreover the disk algebra a d and more in general any uniform algebra whose choquet boundary has no isolated points is proved to have the polynomial daugavet property most of those properties are extended to the vector valued version a x of a uniform algebra a

appendix to the paper some uniserial representations of certain special linear groups by p sin and j g thompson abstract this appendix collects some material on the cohomology of discrete subgroups of lie groups and the theory of their representation varieties to provide background for the results of sin and thompson 2010 2013 in 22 23

on locally indicable groups a group is said to be locally indicable if each of its non trivial finitely generated subgroups has the infinite cyclic group as a homomorphic image such groups were studied by higman 9 in connection with the zero divisor and unit problems for group rings more recently they have arisen i 2 12 in the study of equations over groups in 12 i proved a freiheitsatz for locally indicable groups this has been proved independently by brodskii 2 and short 22 the present paper arises from an investigation of a question put to me by s j pride whether torsionfree 1 relator groups are locally indicable the question was raised originally by baumslag i 1 problem 19 and an affirmative solution has recently been announced by brodski 2 as a consequence the group algebra rg of a torsion free 1 relator group g over an integral domain r has no non trivial zero divisors and no non trivial units using higman s results 9 the first fact was also proved by lewin and lewin 16 by embedding rg in a division ring the second appears to be new as was pointed out to me by k a brown a second consequence is that no 1 relator group has a non trivial finitely generated perfect subgroup which answers 1 problem 7 and 10 question 1 in fact using the freiheitsatz and the tower method described in 12 it is possible to prove the following general version of brodskii s theorem

on the asphericity of ribbon disc complements the complement of a ribbon n disc in the 71 2 ball has a 2 dimensional spine which shares some of the combinatorial properties of classical knot complement spines it is an open question whether such 2 complexes are always aspherical to any ribbon disc we associate a labelled oriented tree from which the homotopy type of the complement can be recovered and we prove asphericity in certain special cases described by conditions on this tree our main result is that the complement is aspherical whenever the associated tree has diameter at most 3

branched coverings of 2 complexes and diagrammatic reducibility the condition that all spherical diagrams in a 2 complex be reducible is shown to be equivalent to the condition that all finite branched covers be aspherical this result is related to the study of equations over groups furthermore large classes of 2 complexes are shown to be diagrammatically reducible in the above sense in particular every 2 complex has a subdivision which admits a finite branched cover which is diagrammatically reducible a consequence of the classical riemann hurwitz formula for riemann surfaces 6 p 301 is the fact that a finite branched cover of an aspherical riemann surface is also aspherical however the property of asphericity of a 2 complex implying asphericity of finite branched covers fails for general 2 complexes in fact we shall prove theorem 4 5 below that this property holds precisely for those 2 complexes which are diagrammatically reducible a 2 complex x is said to be diagrammatically reducible abbreviated dr x if for every combinatorial map f s2 x of the 2 sphere into x a spherical diagram in x there is a pair of faces in the domain with an edge in common which are mapped mirrorwise by f across that edge a closely related notion was introduced by lyndon and schupp 13 but this notion appears in its present form in a paper by sieradski 14 in that same paper sieradski asked a question which would have implied that all classical knot complements in the 3 sphere are homotopy equivalent to diagrammatically reducible 2 complexes we shall prove this latter assertion theorem 6 5 making use of our amalgamation theorem for diagrammatic reducibility theorem 5 4 the characterization of diagrammatic reducibility in terms of branched covers llows from a lifting theorem which may be of general interest if f x y is a combinatorial map of combinatorial 2 complexes then f induces a map of the link complexes of vertices lf lx ly the star graphs or coinitial graph in group theoretical language the map f is said to be reduced if lf is an immersion we prove theorem 4 2 th t if f x y is a reduced map of 2 complexes with x finite then there is a commutative diagram see below where qt z y is a finite branched cover and where g is injective off the zero skeleton of x this result is received by the editors july 9 1986 1980 muthenluticssulviectclussificwtios 1985 retvisios primary20f05 57m12 secondary57m20 1987 american mathematical scoclety 0002 9947 87 1 0 25 pel page

on the dynamics and the fixed subgroup of a free group automorphism on considere des automorphismes f n f n ou f n f a 1 a n est le groupe libre de base a 1 a 2 a n n 2 on compose un champ vectoriel sur un graphe d on introduit des aretes repulsives attractives et des points fixes a l infini et on etudie la dynamique resultante

banach spaces with the daugavet property a banach space x is said to have the daugavet property if every operator t x x of rank 1 satisfies 11 id tl 1 fltil we show that then every weakly compact operator satisfies this equation as well and that x contains a copy of t1 however x need not contain a copy of l1 we also study pairs of spaces x c y and operators t x y satisfying i i j t i i 1 4 1f t i i where j x y is the natural embedding this leads to the result that a banach space with the daugavet property does not embed into a space with an unconditional basis in another direction we investigate spaces where the set of operators with f id tll 1 il t l i is as small as possible and give characterisations in terms of a smoothness condition

detection of essential surfaces in 3 manifolds with sl 2 trees according to bass serre theory se given a field f with a discrete valuation v f z there is a canonical way to construct a simplicial tree t tf v on which sl2 f acts simplicially without inversion when a group has a representation into the group sl2 f then there is an induced action of on the tree via the representation if further is the fundamental group of a compact 3 manifoldm and the action of on the tree t is nontrivial meaning that there is no point on t fixed by every element of then the action induces a splitting of m along an essential surface in the following way let m p m be the universal covering there is a 1 m equivariant map f m t which is transverse to the set e of midpoints of edges in t such that p f 1 e is an essential surface in m we say such an essential surface is associated to or detected by an sl2 tree a natural question is which essential surfaces in a compact 3 manifold can be associated to sl2 trees and how do they depend on the choice of the field and discrete valuation this is the main issue we are going to address in this paper any 3 manifold mentioned in this paper is automatically assumed orientable and connected by an essential surface in a compact 3 manifold m we mean an orientable properly embedded incompressible surface each component of which is neither boundary parallel in m nor bounds a 3 ball when it is a 2sphere also recall that a discrete valuation on a field f is a homomorphism v from the multiplicative group f onto the group of integers z such that v a b min v a v b for any a b f with a b 0 let v f z be a discrete valuation the set ov 0 a f v a 0 s schanuel x zhang department of mathematics suny at buffalo buffalo ny 14260 2900 usa e mail schanuel buffalo edu xinzhang math buffalo edu