Five papers have been published by Geometry & Topology
(4) Geometry & Topology 16 (2012) 1-110 Ideal boundaries of pseudo-Anosov flows and uniform convergence groups with connections and applications to large scale geometry by Sergio Fenley URL: http://www.msp.warwick.ac.uk/gt/2012/16-01/p001.xhtml DOI: 10.2140/gt.2012.16.1
(6) Geometry & Topology 16 (2012) 127-154 Quilted Floer trajectories with constant components: Corrigendum to the article ``Quilted Floer cohomology'' by Katrin Wehrheim and Chris T Woodward URL: http://www.msp.warwick.ac.uk/gt/2012/16-01/p0C1.xhtml DOI: 10.2140/gt.2012.16.127
(1) Statistical hyperbolicity in groups by Moon Duchin, Samuel Lelievre and Christopher Mooney
In this paper, we introduce a geometric statistic called the `sprawl' of a group with respect to a generating set, based on the average distance in the word metric between pairs of words of equal length. The sprawl quantifies a certain obstruction to hyperbolicity. Group presentations with maximum sprawl (ie without this obstruction) are called `statistically hyperbolic'. We first relate sprawl to curvature and show that nonelementary hyperbolic groups are statistically hyperbolic, then give some results for products and for certain solvable groups. In free abelian groups, the word metrics are asymptotic to norms induced by convex polytopes, causing several kinds of group invariants to reduce to problems in convex geometry. We present some calculations and conjectures concerning the values taken by the sprawl statistic for the group Z^d as the generators vary, by studying the space R^d with various norms.
(2) A lower bound for the number of group actions on a compact Riemann surface by James W Anderson and Aaron Wootton
We prove that the number of distinct group actions on compact Riemann surfaces of a fixed genus sigma >= 2 is at least quadratic in sigma. We do this through the introduction of a coarse signature space, the space K_sigma of "skeletal signatures" of group actions on compact Riemann surfaces of genus sigma. We discuss the basic properties of K_sigma and present a full conjectural description.
(3) Expanders and property A by Ana Khukhro and Nick J Wright
We give a cohomological characterisation of expander graphs, and use it to give a direct proof that expander graphs do not have Yu's property A.
(4) Ideal boundaries of pseudo-Anosov flows and uniform convergence groups with connections and applications to large scale geometry by Sergio Fenley
Given a general pseudo-Anosov flow in a closed three manifold, the orbit space of the lifted flow to the universal cover is homeomorphic to an open disk. We construct a natural compactification of this orbit space with an ideal circle boundary. If there are no perfect fits between stable and unstable leaves and the flow is not topologically conjugate to a suspension Anosov flow, we then show: The ideal circle of the orbit space has a natural quotient space which is a sphere. This sphere is a dynamical systems ideal boundary for a compactification of the universal cover of the manifold. The main result is that the fundamental group acts on the flow ideal boundary as a uniform convergence group. Using a theorem of Bowditch, this yields a proof that the fundamental group of the manifold is Gromov hyperbolic and it shows that the action of the fundamental group on the flow ideal boundary is conjugate to the action on the Gromov ideal boundary. This gives an entirely new proof that the fundamental group of a closed, atoroidal $3$--manifold which fibers over the circle is Gromov hyperbolic. In addition with further geometric analysis, the main result also implies that pseudo-Anosov flows without perfect fits are quasigeodesic flows and that the stable/unstable foliations of these flows are quasi-isometric foliations. Finally we apply these results to (nonsingular) foliations: if a foliation is R-covered or with one sided branching in an aspherical, atoroidal three manifold then the results above imply that the leaves of the foliation in the universal cover extend continuously to the sphere at infinity.
(5) Small generating sets for the Torelli group by Andrew Putman
Proving a conjecture of Dennis Johnson, we show that the Torelli subgroup I_g of the genus g mapping class group has a finite generating set whose size grows cubically with respect to g. Our main tool is a new space called the handle graph on which I_g acts cocompactly.
(6) Quilted Floer trajectories with constant components: Corrigendum to the article ``Quilted Floer cohomology'' by Katrin Wehrheim and Chris T Woodward
We fill a gap in the proof of the transversality result for quilted Floer trajectories in [Geom. Topol. 14 (2010) 833--902] by addressing trajectories for which some but not all components are constant. Namely we show that for generic sets of split Hamiltonian perturbations and split almost complex structures, the moduli spaces of parametrized quilted Floer trajectories of a given index are smooth of expected dimension. An additional benefit of the generic split Hamiltonian perturbations is that they perturb the given cyclic Lagrangian correspondence such that any geometric composition of its factors is transverse and hence immersed.
(7) Generalized Monodromy Conjecture in dimension two by Andres Nemethi and Willem Veys
The aim of the article is an extension of the Monodromy Conjecture of Denef and Loeser in dimension two, incorporating zeta functions with differential forms and targeting *all* monodromy eigenvalues, and also considering singular ambient spaces. That is, we treat in a conceptual unity the poles of the (generalized) topological zeta function and the monodromy eigenvalues associated with an analytic germ f:(X,0)->(C,0) defined on a normal surface singularity (X,0). The article targets the "right" extension in the case when the link of (X,0) is a homology sphere. As a first step, we prove a splice decomposition formula for the topological zeta function Z(f,omega;s) for any f and analytic differential form omega, which will play the key technical localization tool in the later definitions and proofs.
Then, we define a set of "allowed" differential forms via a local restriction along each splice component. For plane curves we show the following three guiding properties: (1) if s_0 is any pole of Z(f,omega;s) with omega allowed, then exp(2*pi*i*s_0) is a monodromy eigenvalue of f, (2) the "standard" form is allowed, (3) every monodromy eigenvalue of f is obtained as in (1) for some allowed omega and some s_0.
For general (X,0) we prove (1) unconditionally, and (2)--(3) under an additional (necessary) assumption, which generalizes the semigroup condition of Neumann--Wahl. Several examples illustrate the definitions and support the basic assumptions.
(8) Krull dimension for limit groups by Larsen Louder
We show that varieties defined over free groups have finite Krull dimension, answering a question of Z Sela.
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