Splittings of finitely generated groups over two-ended subgroups

We describe a means of constructing splittings of a one-ended finitely generated group over two-ended subgroups, starting with a finite collection of codimension-one two-ended subgroups. In the case where all the two-ended subgroups have two-ended commensurators, we obtain an annulus theorem, and a form of the JSJ splitting of Rips and Sela. The construction uses ideas from the work of Dunwoody, Sageev and Swenson. We use a particular kind of order structure which combines cyclic orders and treelike structures. In the special case of hyperbolic groups, this provides a link between combinarorial constructions, and constructions arising from the topological structure of the boundary. In this context, we recover the annulus theorem of Scott and Swarup. We also show that a one-ended finitely generated group which contains an infinite-order element, and such that every infinite cyclic subgroup is (virtually) codimension-one is a virtual surface group.

     

Atoroidal Surface Bundles Over Surfaces

A surface-by-surface group is an extension of a non-trivial orientable closed surface group by another such group. It is an open question as to whether every such group contains a free abelian subgroup of rank 2. We show that, for given base and fibre genera, all but finitely many isomorphism classes of surface-by-surface group contain such an abelian subgroup. This can be rephrased in terms of atoroidal surface bundles over surfaces, or in terms of purely loxodromic surface subgroups of the mapping class groups.     

The Cannon–Thurston map for punctured-surface groups

Let Γ be the fundamental group of a compact surface group with non-empty boundary. We suppose that Γ admits a properly discontinuous strictly type preserving action on hyperbolic 3-space such that there is a positive lower bound on the translation lengths of loxodromic elements. We describe the Cannon–Thurston map in this case. In particular, we show that there is a continuous equivariant map of the circle to the boundary of hyperbolic 3-space, where the action on the circle is obtained by taking any finite-area complete hyperbolic structure on the surface, and lifting to the boundary of hyperbolic 2-space. We deduce that the limit set is locally connected, hence a dentrite in the singly degenerate case. Moreover, we show that the Cannon–Thurston map can be described topologically as the quotient of the circle by the equivalence relations arising from the ends of the quotient 3-manifold. For closed surface bundles over the circle, this was obtained by Cannon and Thurston. Some generalisations and variations have been obtained by Minsky, Mitra, Alperin, Dicks, Porti, McMullen and Cannon. We deduce that a finitely generated kleinian group with a positive lower bound on the translation lengths of loxodromics has a locally connected limit set assuming it is connected.     

Hausdorff dimension and dendritic limit sets

Let be a singly degenerate closed surface group acting properly discontinuously on hyperbolic 3-space, H³, such that H³/ has positive injectivity radius. It is known that the limit set is a dendrite of Hausdorff dimension 2. We show that the cut-point set of the limit set has Hausdorff dimension strictly less than 2.     

Central limit theorems for biased randomly trapped random walks on Z

We prove CLTs for biased randomly trapped random walks in one dimension. By considering a sequence of regeneration times, we will establish an annealed invariance principle under a second moment condition on the trapping times. In the quenched setting, an environment dependent centring is necessary to achieve a central limit theorem. We determine a suitable expression for this centring. As our main motivation, we apply these results to biased walks on subcritical Galton–Watson trees conditioned to survive for a range of bias values.     

A topological characterisation of hyperbolic groups

We characterise word hyperbolic groups as those groups which act properly discontinuously and cocompactly on the space of distinct triples of a compact metrisable space. This is, in turn, equivalent to a convergence group for which every point of the space is a conical limit point.

     

A Variation on the Unique Product Property

A variation on the unique product property of groups is describedwhich seems natural from a geometric point of view. It is strongerthan the unique product property, and hence implies, for example,that the group rings have no zero divisors. Some of its closureproperties are described. It is shown that most surface groupssatisfy this condition, and various other examples are given.It is explained how these ideas can give a more geometric interpretationof Promislow's example of a non-unique-product group.     

Length Bounds on Curves Arising from Tight Geodesics

Let M be a complete hyperbolic 3-manifold admitting a homotopy equivalence to a compact surface ∑, such that the cusps of M are in bijective correspondence with the boundary components of ∑. Suppose we realise a tight geodesic in the curve complex as a sequence of closed geodesics M. There is an upper bound on the lengths of such curves in terms of the lengths of the terminal curves and the topologicial type of ∑. We give proofs of these and related bounds. Similar bounds have been proven by Minsky using the sophisticated machinery of hierarchies. Such bounds feature in the work of Brock, Canary and Minsky towards the ending lamination conjecture, and can also be used to study the action of the mapping class group on the curve complex. Received: January 2006, Revision: March 2007, Accepted: July 2007     

Continuously many quasiisometry classes of 2-generator groups

We construct continuously many quasiisometry classes of torsion-free 2-generator small cancellation groups. Received: January 5, 1997