Speaker: Gilbert Baumslag

Title: Reflections on algebraic geometry over groups

Abstract: The object of this talk is to put in place the framework of a theory for groups which parallels the beginnings of classical algebraic geometry and to reflect on analogues in this theory of the Hilbert basis theorem and Hilbert's nullstellenssatz. The work that will be described is joint with Alexei Miasnikov and Vladimir Remeslennikov.

Speaker: Robert Bieri

Title: Openness results for group actions on metric spaces

Abstract: I will report on joint work with Ross Geoghegan aiming towards extending scope and range of application of the Bieri-Neumann-Strebel-Renz--geometric invariant.

Let G be a group of type F_n and X the universal cover of a K(G,1)-complex with finite n-skeleton. Central to our techniques is the concept of ``finitary map'' f: X^n -> X^n -- a generalization of G-equivariant map. We consider a locally compact geodesic metric space M and the space R(G,M) of all actions of G on M by isometries. Every action r in R(G,M) can be lifted to a G-equivariant map h: X -> M and we use this lift h to define whether our finitary map f: X^n -> X^n `` shifts in the direction'' of a point a in M (or, for that matter, in the direction of an end e in bd M). Then we prove, as our main application, that the set of all representations r in R(G,M) for which such a finitary shift f: X^n -> X^n towards some a in M exists is an open subset of R(G,M). In the case when r in R(G,M)is a discrete representation it turns out that the existence of a finitary shift f as above is equivalent to the property that the point stabilizer Stab_G(a) is of type F_n. Thus our general result contains an openness result on stabilizers of such actions. The special case when n = 1 and M is the Euclidean space E^k with G acting by translations recovers Walter Neumann's openness result (1979) on the set of all finitely generated normal subgroups with infinite cylic quotients, which was one of the starting points of the geometric invariant.

Speaker: Brian Bowditch

Title: Connectedness properties of ideal boundaries and limit sets

Abstract: We describe some results relating to convergence actions on continua which have applications to boundaries of hyperbolic groups and limit sets of geometrically finite groups. For example, under certain hypotheses one can deduce that every global cut point is a parabolic fixed point. In particular one can use this in showing that the boundary of a one-ended hyperbolic group is locally connected. Along similar lines one can derive the JSJ splitting of such a group. The techniques involve ideas from convergence groups, actions on R-trees and the theory of group splittings.

Speaker: Martin Bridson

Title: Groups acting on non-positively curved complexes

Abstract: Following a brief survey of what is known about groups which act on spaces of non-positive curvature, I shall concentrate on the case of 2-complexes and explain how the geometry can be used to obtain results of group theoretic interest: these include the classification of subgroups of FxF and the construction of biautomatic groups with no finite quotients.

Speaker: Jim Cannon

Title: On the Conformality of Planar Tiling-Sequences

Abstract: Joint authors: W. J. Floyd and W. R. Parry. A tiling-sequence is a sequence of tilings of a planar or spherical domain where the first tiling is given and the subsequent tilings arise from the first by a well-defined finite subdivision rule. The fundamental problem is to determine which tiling-sequences, given combinatorially, can be realized geometrically so that they are (essentially) preserved by the action of a Kleinian group. The characteristic property of those that can be so realized is a combinatorial conformal-modulus property that we call the conformality condition. Recognizing conformality is a delicate problem with a beautiful theory. At present we can handle tiling-sequences in which there is a great deal of reflectional symmetry. We can also handle constant valence subdivision rules with only one tile type (for which tilings have natural fractal structure but an underlying linear algebra). Conformality can be interpreted as a problem associated with the dynamics of combinatorially given expanding maps of the plane. We intend to explain the state of the art in recognizing conformality. Our dream is to recognize conformality with enough precision to solve the problem for which the notion was created: is a group Kleinian if it is negatively curved ( = word hyperbolic) and has as its space at infinity the 2-sphere?

Speaker: Ruth Charney

Title: Nonpositively curved piecewise Euclidean structures on hyperbolic manifolds

Abstract: (Joint work with M. Davis and G. Moussong) We show that every hyperbolic Riemannian manifold can be given a piecewise Euclidean structure with "extra large" links. We also discuss a generalization of this "flattening" procedure (due to C-K Wong) to piecewise hyperbolic complexes.

Speaker: Marston Conder

Title:Group actions and regular maps on non-orientable surfaces

Abstract: In this lecture I will show that for every positive integer p there exists a compact non-orientable surface of genus p with automorphism group of order at least 4p if p is odd, or at least 8(p-2) if p is even. The lower bound of 4p represents a doubling of the former known lower bound on the maximum number of symmetries of a non-orientable surface of given genus. I will also report on progress (in joint work with Colin Maclachlan and Steve Wilson) in proving the lower bound of 4p is sharp for infinitely many p congruent to 3 modulo 12. This turns out to be closely related with the question of existence of regular maps (symmetric graph embeddings) on non-orientable surfaces of certain genera, investigated in joint work with Brent Everitt.

Speaker: Daryl Cooper

Title: Actions on rational homology 3-spheres

Abstract: We show that every finite group acts freely on some rational homology 3-sphere. This is joint work with Darren Long.

Speaker: John Crisp

Title: Injectivity of maps between Artin groups

Abstract: I shall describe a sufficient condition for the injectivity of a homomorphism between Artin semigroups which allows one to describe a large class of such injective maps. The homomorphism between Artin groups induced by any map in this class may be topologically realised via a corresponding inclusion of Salvetti complexes. In the case that the Artin groups involved are both of finite type, the group homomorphism is found to be injective and hence the inclusion of complexes is \pi_1-injective.

Speaker: Robert Gilman

Title: A small cancellation characterization of hyperbolic groups

Abstract: The (word) hyperbolic groups of Gromov are defined by a geometric condition on their Cayley diagrams, and small cancellation presentations are a rich source of hyperbolic groups of dimension two. We present a small cancellation condition which characterizes all hyperbolic groups. That is, every hyperbolic group admits a presentation satisfying the condition; and every such presentation defines a word hyperbolic group.

Speaker: Stephen Glasby

Title: Unique tensor factorization of irreducible projective representations of groups

Abstract: A projective representation of a finite group G can always be expressed as an internal tensor product of projective representations of G each of which is `tensor-irreducible'. This talk considers when such a tensor factorization is `unique'. In the negative direction unique tensor factorization can fail to hold even when G is cyclic and the underlying field is algebraically closed. In the positive direction we prove, for certain classes of groups and fields, that unique tensor factorization of irreducible projective representations does hold.

Speaker: Mikhael Gromov

Title: On the group geometry related to the Novikov conjecture

Abstract: (to be submitted)

Speaker: Susan Hermiller

Title: Artin groups, rewriting systems, and three-manifolds

Abstract: In joint work with John Meier, we construct finite complete rewriting systems for two large classes of Artin groups: those of finite type, and those whose defining graphs are based on trees. The constructions in the two cases are quite different; while the construction for Artin groups of finite type uses normal forms introduced through work on complex hyperplane arrangements, the rewriting systems for Artin groups based on trees are constructed via three-manifold topology. This construction naturally leads to the question: Which Artin groups are three-manifold groups? Although we do not have a complete solution, the answer, it seems, is ``not many."

Speaker: John Hudson

Title: Canonical Words for Braids

Abstract: The braid groups are interesting examples of automatic groups that do not appear to be shortlex automatic (except for braids with =BE 3 strands). Thurston's canonical words will normally be much longer than the minimum length equivalent word. Kay Tatsuoka has proposed a canonical form for braid words, based on the representation of the Cayley graph as a union of 'flat' subgraphs. This paper investigates the recognisibility of such canonical forms.

Speaker: Renfang Jiang

Title: Out(F_n)-action on a simplicial complex

Abstract:Let G be a group with actions on trees. The G-trees can be parametrized by the space of the translation length functions. The outer automorphism group Out(G) acts on it. Hatcher and Vogtmann studied the simplicial complex SF_n, which is the (unordered) factorizations Z_0*...*Z_k*H of F_n. They proved that SF_n is ((n-3)/2)-connected. They used SF_n in their calculation of the homology groups of Aut(F_n). We redefine SF_n using our notations. By applying the theory developed by Bass and Jiang, we calculate the vertex stabilizers of Out(F_n)-action on SF_n. A minimal Out(F_n)-invariant subgraph, Y, of the one skeleton of SF_n is defined. The vertices of Y are the equivalence classes of F_n-trees whose quotient graph is either an n-leaf rose or a (n-1)-leaf rose. We obtain the following exact sequence 1->\pi_1(Y)->G_0*_{G_{01}}G_1 ->Out(F_n)->1, where G_0 is the automorphism group of an n-leaf rose, and G_1 is virtually abelian.

Speaker: Bill Kantor

Title: Reconstructing classical group actions

Abstract: Suppose that a finite group G, given as a permutation group, a matrix group or just a black box group, is known to be isomorphic to a classical simple group. This talk concerns algorithms for reconstructing a natural classical group projective module for G.

Speaker: Ann Chi Kim

Title: A group presentation and 3-dimensional manifolds

Abstract: We shall consider a tessellation on the 2-sphere consisting of two m-gons and 2n pentagons. Identifying pairs of faces, we shall get a closed, connected and orientable 3-dimensional manifold. From the construction we shall show that the manifolds will be the 2-fold coverings of S^3 over some link.

Speaker: Charles Leedham-Green

Title: Structure and classification of p-groups and pro-p-groups

Abstract: (to be submitted)

Speaker: Gus Lehrer

Title: Invariant hypersurfaces and reflection groups

Abstract: (to be submitted)

Speaker: Ian G. Macdonald

Title: (to be announced)

Abstract: (to be submitted)

Speaker: Gaven Martin

Title: The Geometry of Kleinian Groups

Abstract: We show how a new family of polynomial trace identities can be used to describle the space of all two generator discrete subgroups of PSL(2,C). Application of this description to the theory of Kleinain groups and Hyperbolic Geometry are then discussed.

Speaker: Saburo Matsumoto

Title: Subgroup Separability of 3-Manifold Groups

Abstract: I will mention an example (due to Burns, Karrass, and Solitar) of a compact 3-manifold M whose fundamental group is not subgroup separable (non-LERF) and show that M is a graph manifold and its fundamental group is an HNN extension. Using a method of Rubinstein and Wang, I will show that some surfaces (with boundary) immersed in M cannot lift to an embedding in any finite-degree cover of M. I will also mention some extensions of this criterion. Finally, I will construct 3-manifolds with a cubing of non-positive curvature which admit non-separable immersed surfaces and conclude the talk with proposing some related problems and conjectures.

Speaker: Lee Mosher

Title: Quasi-isometry classification of the solvable Baumslag-Solitar groups

Abstract: Joint with Benson Farb. Abstract: The solvable Baumslag-Solitar groups are the groups BS(1,n) = < a,b : b a b^{-1} = a^n > It is well known that BS(1,n) and BS(1,m) are commensurable if and only if m,n are powers of a common integer. We prove that they are quasi-isometric if and only if they are commensurable. The proof uses a metric 2-complex X_n on which BS(1,n) acts properly discontinuously and cocompactly. We define an "upper" and "lower" boundary of X_n, and we put metrics on these boundaries, so that the bilipschitz classes of these metric spaces are quasi-isometry invariants of BS(1,n). The upper boundary is the space of hyperbolic planes in X_n, and it is isometric to the n-adic rational numbers Q_n. The proof is finished by applying a theorem of Daryl Cooper which says that Q_n and Q_m are bilipschitz equivalent if and only if n,m are powers of a common integer. We also discuss some progress towards characterizing those finitely generated groups which are quasi-isometric to BS(1,n).

Speaker: Walter Neumann

Title:Biautomaticity of central extensions and bounded cohomology

Abstract: The talk will mostly be on joint work with Lawrence Reeves in which we showed that any virtually central extension of a word hyperbolic group is biautomatic. The proof raises new questions about bounded cohomology and related concepts. In particular, it has been used in recent work of S. Gersten on l_\infty cohomology which will also be briefly described.

Speaker: Mike Newman

Title: Updating the Burnside order question

Abstract: In his paper of 1902 Burnside asked three questions about periodic groups. In the light of progress on the first two and the related Restricted Burnside Problem, it seems appropriate to update the third.

Speaker: Viatcheslav N. Obraztsov

Title: Some new embedding constructions

Abstract: Some new embedding schemes of arbitrary sets of groups into infinite groups with preassigned properties have been obtained using the method of cancellation diagrams. Among their applications we mention the existence of the following groups: 1) a 2-generator simple infinite complete group all of whose maximal proper subgroups are infinite dihedral; 2) an uncountable group G in which all proper subgroups are countable such that OutG is a given group of cardinality less than or equal to |G| and G contains every countable group (assuming CH); 3) a nondiscrete Hausdorff topological group of cardinality $\aleph_{n}$, where n is any nonnegative integer, with no proper subgroups of the same cardinality and with each proper subgroup discrete; 4) a group of a given infinite cardinality which is isomorphic to all its nontrivial normal subgroups and other than simple groups, free groups of infinite rank and the infinite cyclic group.

Speaker: Eamonn O'Brien

Title: Tensor factorisations of matrix groups

Abstract:Given a finite dimensional vector space V, we construct a family of projective geometries whose flats are certain subspaces of V, and show that there is a one-to-one correspondence between this family of projective geometries and the set of equivalence classes of tensor decompositions of V. This provides one component of a practical algorithm for finding a tensor decomposition of a finite dimensional KG-module, or proving that no non-trivial tensor decomposition of the module exists. We will discuss these geometries and outline aspects of the algorithm.

Speaker: A.Yu.Olshanskii

Title: Locally finite subgroups of free Burnside groups of large even exponents

Abstract: The main part of the talk will present recent joint results of S.V.Ivanov and A.Yu.Olshanskii on locally finite subgroups of free Burnside group B(m,n) of even exponent n. We consider n \ge 2^48, n is divisible by 2^9 and the number of generators m > 1. Under these assumptions, a complete description of all infinite groups that are embeddable in B(m,n) as (maximal) locally finite subgroups is given. Any infinite locally finite subgroup L of B(m,n) is contained in unique maximal locally finite subgroup while any finite 2- subgroup of B(m,n) is contained in continuously many non-isomophic maximal locally finite subgroups. In addition, L is locally conjugate to a maximal locally finite subgroup of B(m,n). To prove these and other results, centralizers of subgroups in B(m,n) are investigated. For example, it is proven that the centralizer of a finite 2-subgroup of B(m,n) contains a subgroup isomorphic to a free Burnside group of countably infinite rank and exponent n; the centralizer of a finite non-2-subgroup of B(m,n) or the centralizer of a non-locally finite subgroup of B(m,n) is always finite; the centralizer of a subgroup S is infinite if and only if S is a locally finite

Speaker: Christophe Pittet

Title: Isoperimetric profiles of solvable groups

Abstract: On a finitely generated group, the isoperimetric profile is an asymptotic invariant which is strongly related to random walks. We will explain how to compute it for some solvable groups.

Speaker: Cheryl Praeger

Title: The number of fixed-point-free orthogonal transformations of finite vector spaces of even order

Abstract: I will discuss the problem of estimating the number of nonsingular matrices in the orthogonal group $O^\pm (d,q)$ ($d$ and $q$ even) for which the only fixed vector in the underlying vector space $V$ is the zero vector. This problem arose in connection with a more general study by Peter M.~Neumann and myself of the proportion of cyclic matrices in classical groups. The techniques we had developed failed (spectacularly) to give a usable lower bound precisely in the case of orthogonal groups over fields of order $2$.

In this exceptional case we derived a recursion relation between the number of unipotent matrices and the number of fixed-point-free matrices in orthogonal groups of various dimensions. This allowed us to derive a functional relation between the generating functions for these quantities. From these generating functions, and a result of Steinberg about the numbers of unipotent matrices, we were able to obtain the bounds we needed.

Speaker: Steve Pride

Title: The geometry of group extensions

Abstract: If G is a group then by a 3-presentation (for G) we will mean a pair (P, D) where P is an ordinary presentation of G and D is a collection of spherical diagrams (or pictures) over P representing a set of generators of the second homotopy module of P. A basic question is the following: if G is an extension of K by H, and if one knows 3-presentations for K and H, then determine a 3-presentation for G. This poblem has been recently solved in joint work with Y.G. Baik and J. Harlander. In fact, this work considers wider questions. Let P(H) = < A; R, > P(K) = < X; T > be presentations for H, K respectively. Then any extension of K by H will have a presentation of the form (*) P = < A, X; r=w(r) (r in R), T, xa=au(a,x), ax=v(a,x)a (a in A, x in X) > where the w's, u's and v's are words on X. However, in general, if G is the group defined by P then G will not be an extension of K by H because the natural map K--->G will not be injective. We will say that P is developable if this map is injective. By analysing presentations of the form (*) geometrically we show that there is a procedure (which can be made algorithmic under suitable conditions on P(H), P(K)) to determine for any P as in (*) whether or not it is developable, and if so to compute a 3-presentation for G given 3-presenations for H and K. When P is not developable one can obtain a presentation on X for the subgroup of G generated by X. (A special case of this was done by Conway/Coxeter/Shephard.). Morover, one can still compute a 3-presentation for G.

Speaker: Zlil Sela

Title: Moduli spaces of residually free groups

Abstract:(to be submitted)

Speaker: Mike Shapiro

Title: Almost a generalization of a theorem of Muller and Schupps

Abstract: A number of theorems connect classes of groups with the sorts of computing machines necessary to solve the word problem in those groups. For example, Anissimov and Seifert show that a group is finite if and only if its word problem can be solved by a finite state automaton. A celebrated theorem of Muller and Schupp states that the word problem in a group can be solved by means of a pushdown automaton if and only if that group is virtually free. There is a class of computing machines which generalize the pushdown automata, to wit: nested stack automata. These machines have their storage capacity organized in a tree-like manner. This increase in computing power translates into an ability to recognize considerably more complicated languages. Surprisingly, this increased computing power does not seem to confer any group-theoretic advantage. In joint work with Gilman and Grigorchuk, we show that under mild assumptions on our nested stack automaton, the word problem in a finitely presented group can be solved by means of a nested stack automaton if and only if that group is virtually free.

Speaker: Hamish Short

Title: Isoperimetric inequalities for subgroups of HNN extensions

Abstract: When G is an HNN extension with finitely presented base group H, we establish an isoperimetric inequality for H in terms of the isoperimetric inequality satisfied by G. In particular, we show that Noel Brady's example of a non word hyperbolic finitely presented subgroup of a word hyperbolic group satisfies a polynomial isoperimetric inequality.

We shall also discuss some joint work with Thomas Delzant concerning listing all word hyperbolic groups.

Speaker: Steve Smith

Title: Applications of finite geometries to group cohomology

Abstract: The talk will survey recent developments: Recently group cohomologists have become interested in the sporadic simple groups. The study of certain building-like geometries for these groups seems to contribute useful information for the cohomology, for example ways of checking stability of cohomology of subgroups, and Webb-type formulas for cohomology as an alternating sum of that of certain subgroups.

Speaker: John Stallings

Title: Cut Vertices and Waves

Abstract:Given a finitely generated free group F with a basis A, and a finite set of elements R of F, there exists an algorithm to determine whether or not some set of conjugates of the elements of R belongs to a proper free factor of F. This can be done, in fact, by using the ``cut vertex'' idea of Whitehead from 1936. Geometrically, this can be mirrored on a Heegaard diagram of a closed orientable 3-manifold, and the result is to obtain what Volodin, Kuznetsov, and Fomenko in 1974 called a ``wave'', for certain special types of diagrams; this gives a way to discuss ``strongly irreducible'' diagrams.

Speaker: G.A.Swarup (= A. S. Gadde)

Title: Tracks and splittings of groups

Abstract: Using Dunwoody's tracks, we (G.A.S and G.P.Scott) prove: If G is a torsion free hyperbolic group with one end and if G has a two ended subgroup H such that the number ends of the pair (G,H) is greater than or equal to two, then G splits over some two ended subgroup. Subsequently, the theorem without the assumption about torsion has been proved by B.Bowditch by different methods.

Speaker: George Willis

Title: Totally Disconnected, Locally Compact Groups

Abstract: Each totally disconnected, locally compact group G has a compact, open subgroup U. If U is normal, then G is an extension of the discrete group G/U by the compact group U and this extension describes the relationship between the algebraic and topological structures of G. However, not every totally disconnected, locally compact group has a compact, open, normal subgroup. The talk will discuss some theorems which say precisely how the compact, subgroups of G may fail to be normal. The example of the group of automorphisms of a homogeneous tree will be used to illustrate the theorems. These theorems are strong enough to have answered several conjectures but at the same time suggest further questions and thus appear to point the way to a more complete structure theory for totally disconnected groups.