curvature and rank of teichmuller space let 5 be a surface with genus g and n boundary components and let d s 3g 3 n denote the number of curves in any pants decomposition of s we employ metric properties of the graph of pants decompositions p s to prove that the weil petersson metric on teichmiiller space teich s is gromov hyperbolic if and only if d s 3 the weil petersson metric has higher rank in the sense of gromov it admits a quasi isometric embedding of rk k 2 when d s 3 and that no complete riemannian metric of pinched negative curvature exists on moduli space m s when d s 2 1 introduction the weil petersson metric on teichmiiller space teich 5 has many curious properties it is a riemannian metric with negative sectional curvature but its curvatures are not bounded away from zero or negative infinity it is geodesically convex but it is not complete in this paper we show that in spite of exhibiting negative curvature behavior the weil petersson metric is not coarsely negatively curved except for topologically simple surfaces s our main theorem answers a question of bowditch be question 11 4

the weil petersson hessian of length on teichmuller space we present a brief but nearly self contained proof of a formula for the weil petersson hessian of the geodesic length of a closed curve either simple or not simple on a hyperbolic surface the formula is the sum of the integrals of two naturally defined positive functions over the geodesic proving convexity of this functional over teichmuller space due to wolpert 1987 we then estimate this hessian from below in terms of local quantities and distance along the geodesic the formula extends to proper arcs on punctured hyperbolic surfaces and the estimate to laminations wolpert s result that the thurston metric is a multiple of the weil petersson metric directly follows on taking a limit of the formula over an appropriate sequence of curves we give further applications to upper bounds of the hessian especially near pinching loci recover through a geometric argument wolpert s result on the convexity of length to the half power and give a lower bound for growth of length in terms of twist

deviation inequalities for random walks we study random walks on groups with the feature that roughly speaking successive positions of the walk tend to be aligned we formalize and quantify this property by means of the notion of deviation inequalities we show that deviation inequalities have several consequences including central limit theorems the local lipschitz continuity of the rate of escape and entropy as well as linear upper and lower bounds on the variance of the distance of the position of the walk from its initial point in the second part of this article we show that the exponential deviation inequality holds for measures with exponential tail on acylindrically hyperbolic groups these include nonelementary relatively hyperbolic groups mapping class groups many groups acting on cat 0 spaces and small cancellation groups

effective distance between nested margulis tubes we give sharp effective bounds on the distance between tori of fixed injectivityradius inside a margulis tube in a hyperbolic 3 manifold

median structures on asymptotic cones and homomorphisms into mapping class groups the main goal of this paper is a detailed study of asymptotic cones of the mapping class groups in particular we prove that every asymptotic cone of a mapping class group has a bi lipschitz equivariant embedding into a product of real trees sending limits of hierarchy paths onto geodesics and with image a median subspace one of the applications is that a group with kazhdan s property t can have only finitely many pairwise non conjugate homomorphisms into a mapping class group we also give a new proof of the rank conjecture of brock and farb previously proved by behrstock and minsky and independently by hamenstaedt

uniform bounds for weil petersson curvatures we find bounds for weil petersson holomorphic sectional curvature and the weil petersson curvature operator in several regimes that do not depend on the topology of the underlying surface among other results we show that the minimal weil petersson holomorphic sectional curvature of a sufficiently thick hyperbolic surface is comparable to 1 independently of the genus this provides a counterexample to some suggestions that the weil petersson metric becomes asymptotically flat as the genus g goes to infinity in some loci in teichmuller space adopting a different perspective on curvature we also show that the minimal most negative eigenvalue of the curvature operator at any point in the teichmuller space teich sg of a closed surface sg of genus g is uniformly bounded away from zero restricting to a thick part of teich sg we show that the minimal eigenvalue is uniformly bounded below by an explicit constant which does not depend on the topology of the surface but only on the given bound on injectivity radius

divergence and quasimorphisms of right angled artin groups we give a group theoretic characterization of geodesics with superlinear divergence in the cayley graph of a right angled artin group a with connected defining graph we use this to prove that the divergence of a is linear if is a join and quadratic otherwise as an application we give a complete description of the cut points in any asymptotic cone of a we also show that every non abelian subgroup of a has an infinite dimensional space of non trivial quasimorphisms

cusp geometry of fibered 3 manifolds let f be a surface and suppose that varphi colon f to f is a pseudo anosov homeomorphism fixing a puncture p of f the mapping torus m m varphi is hyperbolic and contains a maximal cusp c about the puncture p we show that the area and height of the cusp torus partial c is equal to the stable translation distance of varphi acting on the arc complex scr a f p up to an explicitly bounded multiplicative error our proof relies on elementary facts about the hyperbolic geometry of pleated surfaces in particular the proof of this theorem does not use any deep results from teichm uller theory kleinian group theory or the coarse geometry of scr a f p a similar result holds for quasi fuchsian manifolds n cong f times bbb r in that setting we find a combinatorial estimate for the area and height of the cusp annulus in the convex core of n up to explicitly bounded multiplicative and additive error as an application we show that covers of punctured surfaces induce quasi isometric embeddings of arc complexes

behavior of geodesic length functions on teichmuller space let t be the teichmuller space of marked genus g n punctured riemann surfaces with its bordification t the augmented teichmuller space of marked riemann surfaces with nodes abi77 ber74 pro vided with the wp metric t is a complete cat 0 metric space dw03 wol03 yam04 an invariant of a marked hyperbolic struc ture is the lengthof the geodesic in a free homotopy class a basic feature of teichmuller theory is the interplay of two dimensional hyperbolic geometry weil petersson wp geometry and the behavior of geodesic length functions our goal is to develop the understanding of the intrinsic local wp geometry through a study of the gradient and hessian of geodesic length functions considerations include ex pansions for the wp pairing of gradients expansions for the hessian and covariant derivative comparability models for the wp metric as well as the behavior of wp geodesics including a description of the alexandrov tangent cone at the augmentation approximations and applications for geodesics close to the augmentation are developed an application for fixed points of group actions is described bound ing configurations and functions on the hyperbolic plane is basic to our approach considerations include analyzing the orbit of a dis crete group of isometries and bounding sums of the inverse square exponential distance

jsj splittings for finitely presented groups over slender groups we generalize the jsj splitting of rips and sela to give decompositions of finitely presented groups which capture splittings over certain classes of small subgroups such classes include the class of all 2 ended groups and the class of all virtually z z groups the approach called track zipping is relatively elementary and differs from the rips sela approach in that it does not rely on the theory of r trees but rather on an understanding of certain embedded 1 complexes called patterns in a presentation 2 complex for the ambient group

dynamics and bifurcations of nonsmooth systems a survey abstract in this survey we discuss current directions of research in the dynamics of nonsmooth systems with emphasis on bifurcation theory an introduction to the state of the art also for non specialists is complemented by a presentation of the main open problems we illustrate the theory by means of elementary examples the main focus is on piecewise smooth systems which have recently attracted a lot of attention but we also briefly discuss other important classes of nonsmooth systems such as nowhere differentiable ones and differential variational inequalities this extended framework allows us to put the diverse range of papers and surveys in this special issue in a common context a dedicated section is devoted to concrete applications that stimulate the development of the field this survey is concluded by an extensive bibliography

quasi convexity of hyperbolically embedded subgroups we show that any infinite order element g of a virtually cyclic hyperbolically embedded subgroup of a group g is morse that is to say any quasi geodesic connecting points in the cyclic group c generated by g stays close to c this answers a question of dahmani guirardel osin what is more we show that hyperbolically embedded subgroups are quasi convex finally we give a definition of what it means for a collection of subspaces of a metric space to be hyperbolically embedded and we show that axes of pseudo anosovs are hyperbolically embedded in teichmuller space endowed with the weil petersson metric

quasi isometry invariance of group splittings we show that a finitely presented one ended group which is not commensurable to a surface group splits over a two ended group if and only if its cayley graph is separated by a quasi line this shows in particular that splittings over two ended groups are preserved by quasi isometries

discontinuity induced bifurcations of equilibria in piecewise smooth and impacting dynamical systems a rich variety of dynamical scenarios has been shown to occur when a fixed point of a non smooth map undergoes a border collision this paper concerns a closely related class of discontinuity induced bifurcations those involving equilibria of n dimensional piecewise smooth flows specifically transitions are studied which occur when a boundary equilibrium that is one lying within a discontinuity manifold is perturbed it is shown that such equilibria can either persist under parameter variations or can disappear giving rise to different bifurcation scenarios conditions to classify among the possible simplest scenarios are given for piecewise smooth continuous filippov and impacting systems also we investigate the possible birth of other attractors e g limit cycles at a boundary equilibrium bifurcation

rank 1 phenomena for mapping class groups we prove that every element of the mapping class group g has linear growth confirming a conjecture of n ivanov and that g is not boundedly generated we also provide restrictions on linear representations of g and its finite index subgroups

non realizability and ending laminations proof of the density conjecture we give a complete proof of the bers sullivan thurston density conjecture in the light of the ending lamination theorem it suffices to prove that any collection of possible ending invariants is realized by some algebraic limit of geometrically finite hyperbolic manifolds the ending invariants are either riemann surfaces or filling laminations supporting masur domain measured laminations and satisfying some mild additional conditions with any set of ending invariants we associate a sequence of geometrically finite hyperbolic manifolds and prove that this sequence has a convergent subsequence we derive the necessary compactness theorem combining the rips machine with non existence results for certain small actions on real trees of free products of surface groups and free groups we prove then that the obtained algebraic limit has the desired conformal boundaries and the property that none of the filling laminations is realized by a pleated surface in order to be able to apply the ending lamination theorem we have to prove that this algebraic limit has the desired topological type and that these non realized laminations are ending laminations that this is the case is the main novel technical result of this paper loosely speaking we show that a filling masur domain lamination which is not realized is an ending lamination

on the hopf bifurcation in control systems with a bounded nonlinearity asymptotically homogeneous at infinity the paper studies existence uniqueness and stability of large amplitude periodic cycles arising in hopf bifurcation at infinity of autonomous control systems with bounded nonlinear feedback we consider systems with functional nonlinearities of landesman lazer type and a class of systems with hysteresis nonlinearities the method is based on the technique of parameter functionalization and methods of monotone concave and convex operators c 2001 academic press

morse theory and finiteness properties of groups we examine the finiteness properties of certain subgroups of right angled artin groups in particular we find an example of a group that is of type fp z but is not finitely presented

quantum morphing and the jones polynomial we will explore the experimental observation that on the set of knots with bounded crossing number algebraically independent vassiliev invariants become correlated as noticed first by s willerton we will see this through the value distribution of the jones polynomial at roots of unit as the degree of the roots of unit is getting larger the higher order fluctuation is diminishing and a more organized shape will emerge from a rather random value distribution of the jones polynomial we call such a phenomenon quantum morphing evaluations of the jones polynomial at roots of unity play a crucial role for example in the volume conjecture

iteration of mapping classes and limits of hyperbolic 3 manifolds let mod s be an element of the mapping class group of a surface s we classify algebraic and geometric limits of sequences q i x y i 1 of quasi fuchsian hyperbolic 3 manifolds ranging in a bers slice when has infinite order with finite order restrictions there is an essential subsurface d s so that the geometric limits have homeomorphism type s d 0 typically has pseudo anosov restrictions and d has components with negative euler characteristic these components correspond to new asymptotically periodic simply degenerate ends of the geometric limit we show there is an s 1 depending on and bounded in terms of s so that q si x y i 1 converges algebraically and geometrically and we give explicit quasi isometric models for the limits

generalized markoff maps and mcshane s identity abstract we study the relative sl 2 c character varieties of the one holed torus and the action of the mapping class group on the relative character variety we show that the subset of characters satisfying two simple conditions called the bowditch q conditions is open in the relative character variety and that the mapping class group acts properly discontinuously on this subset furthermore this is the largest open subset for which this holds we also show that a generalization of mcshane s identity holds for all characters satisfying the bowditch q conditions finally we show that further variations of the mcshane bowditch identity hold for characters which are fixed by an anosov element of the mapping class group and which satisfy a relative version of the bowditch q conditions with applications to identities for incomplete hyperbolic structures on punctured torus bundles over the circle and also for closed hyperbolic 3 manifolds which are obtained by hyperbolic dehn surgery on such manifolds

bennequin s inequality and the positivity of the signature we use an algorithm for special diagrams to prove a bennequin type inequality tor the signature of an arbitrary link diagram related to its murasugi sum decomposition we apply this inequality to show that the signature of a non trivial positive 3 braid knot is greater than its genus and that the signature of a positive braid link is minorated by an increasing function of its negated euler characteristic the latter property is conjectured to extend to positive links

large scale rank of teichmuller space let x be quasi isometric to either the mapping class group equipped with the word metric or to teichmuller space equipped with either the teichmuller metric or the weil petersson metric we introduce a unified approach to study the coarse geometry of these spaces we show that the quasi lipschitz image in x of a box in r is locally near a standard model of a flat in x as a consequence we show that for all these spaces the geometric rank and the topological rank are equal the methods are axiomatic and apply to a larger class of metric spaces

schwarzian derivatives projective structures and the weil petersson gradient flow for renormalized volume to a complex projective structure on a surface thurston associates a locally convex pleated surface we derive bounds on the geometry of both in terms of the norms and 2 of the quadratic differential of given by the schwarzian derivative of the associated locally univalent map we show that these give a unifying approach that generalizes a number of important well known results for convex cocompact hyperbolic structures on 3 manifolds including bounds on the lipschitz constant for the nearest point retraction and the length of the bending lamination we then use these bounds to begin a study of the weil petersson gradient flow of renormalized volume on the space cc n of convex cocompact hyperbolic structures on a compact manifold n with incompressible boundary leading to a proof of the conjecture that the renormalized volume has infimum given by one half the simplicial volume of dn the double of n

rational subgroups of biautomatic groups centralizers of finite subsets in biautomatic groups are them selves biautomatic every polycyclic subgroup of a biautomatic group is abelian by finite

the geometry of the disk complex we give a distance estimate for the disk complex we use the distance estimate to prove that the disk complex is gromov hyperbolic as another application of our techniques we find an algorithm which computes the hempel distance of a heegaard splitting up to an error depending only on the genus

ends of hyperbolic 3 manifolds let n h3 f be a hyperbolic 3 manifold which is homeomorphic to the interior of a compact 3 manifold we prove that n is geometrically tame as a consequence we prove that f s limit set l is either the entire sphere at infinity or has measure zero we also prove that n s geodesic flow is ergodic if and only if lf is the entire sphere at infinity department of mathematics stanford university stanford california 94305 current address department of mathematics university of michigan ann arbor michigan 48109 e mail address dick canary math lsa umich edu this content downloaded from 40 77 167 14 on wed 15 jun 2016 05 36 06 utc all use subject to http about jstor org terms

the effect of frequency and clearance variations on single degree of freedom impact oscillators abstract a simple piecewise linear forced oscillator with impacts is taken as a model for more complicated vibro impacting systems the presence of impacts makes the system discontinuous and non linear and to study it effectively the theory of dynamical systems is used by a combination of analytical and numerical techniques many phenomena are identified which are characteristic of non linear dynamical systems as the system parameters are varied in particular these include chaotic regimes and multiply coexisting stable states the domains of attraction of the periodic states are plotted and compared with the strange attractors of the chaotic states in addition phenomena are observed related to the discontinuity which have no counterpart in smooth dynamical systems chatter when infinitely many impacts occur in a finite time and grazing when a stable periodic orbit encounters a discontinuity and disappears catastrophically under a smooth change in the parameters

the weil petersson metric and volumes of 3 dimensional hyperbolic convex cores we present a coarse interpretation of the weil petersson distance dwp x y between two finite area hyperbolic riemann surfaces x and y using a graph of pants decompositions introduced by hatcher and thurston the combinatorics of the pants graph reveal a connection between riemann surfaces and hyperbolic 3 manifolds conjectured by thurston the volume of the convex core of the quasi fuchsian manifold q x y with x and y in its conformal boundary is comparable to the weil petersson distance dwp x y in applications we relate the weil petersson distance to the hausdorff dimen sion of the limit set and the lowest eigenvalue of the laplacian for q x y and give a new finiteness criterion for geometric limits mathematics department university of chicago 5734 s university ave chicago il 60637 e mail address brock math uchicago edu

hyperbolic structures on 3 manifolds i deformation of acylindrical manifolds this is the first in a series of papers showing that haken manifolds have hyperbolic structures this first was published the second two have existed only in preprint form and later preprints were never completed this eprint is only an approximation to the published version which is the definitive form for part i and is provided for convenience only all references and quotations should be taken from the published version since the theorem numbering is different and not all corrections have been incorporated into the present version parts ii and iii will be made available as eprints shortly

quasi isometric classification of graph manifold groups we show that the fundamental groups of any two closed irreducible nongeometric graph manifolds are quasi isometric we also classify the quasi isometry types of fundamental groups of graph manifolds with boundary in terms of certain finite twocolored graphs a corollary is the quasi isometric classification of artin groups whose presentation graphs are trees in particular any two right angled artin groups whose presentation graphs are trees of diameter greater than 2 are quasi isometric further this quasi isometry class does not include any other right angled artin groups a finitely generated group can be considered geometrically when endowed with a word metric up to quasi isometric equivalence such metrics are unique in this article only finitely generated groups are considered given a collection of groups g gromov 11 proposed the fundamental questions of identifying which groups are quasi isometric to those in g rigidity and which groups in g are quasi isometric to each other classification in this article we focus on the classification question for graph manifold groups and right angled artin groups a compact 3 manifold m is called geometric if m m admits a geometric structure in the sense of thurston i e a complete locally homogeneous riemannian metric of finite volume thurston s geometrization conjecture see 29 17 21 23 22 provides that every irreducible 3 manifold of zero euler characteristic i e with boundary consisting only of tori and klein bottles admits a decomposition along tori and klein bottles into geometric pieces the minimal such decomposition being called the geometric decomposition there is a considerable literature on quasi isometric rigidity and classification of 3 manifold groups the rigidity results can be briefly summarized in the following form

on almost fuchsian manifolds almost fuchsian manifold is a class of complete hyperbolic threemanifolds a such three manifold is a quasi fuchsian manifold that contains a closed incompressible minimal surface with principal curvatures everywhere in the range of 1 1 in such a manifold the minimal surface is unique and embedded hence one can parametrize these hyperbolic three manifolds by their minimal surfaces in this paper we obtain estimates on several geometric and analytical quantities of an almost fuchsian manifold m in terms of the data on the minimal surface in particular we obtain an upper bound for the hyperbolic volume of the convex core of m and an upper bound on the hausdorff dimension of the limit set associated to m we also constructed a quasi fuchsian manifold which admits more than one minimal surface and it does not admit a foliation of closed surfaces of constant mean curvature

a novel route to a hopf bifurcation scenario in switched systems with dead zone abstract planar switched systems with dead zone are analyzed in particular we consider the effects of a perturbation which is applied to a linear control law and due to the perturbation the control changes from purely positional to position velocity control this type of a perturbation leads to a novel hopf like discontinuity induced bifurcation we show that this bifurcation leads to the creation of a small scale limit cycle attractor which scales as the square root of the bifurcation parameter we then investigate numerically a planar switched system with a positional feedback law dead zone and time delay in the switching function using the same parameter values as for the switched system without time delay in the switching function we show a hopf like bifurcation scenario which exhibits a qualitative and a quantitative agreement with the scenario analyzed for the non delayed system

parabolic subgroups of artin groups abstract let a be an artin system for x we denote by a x the subgroup of a generated by x such a group is called a parabolic subgroup of a we reprove van der lek s theorem a parabolic subgroup of an artin group is an artin group we give an algorithm which decides whether two parabolic subgroups of an artin group are conjugate let a be a finite type artin group and let a x be a parabolic subgroup with connected associated coxeter graph the quasi centralizer of a x in a is the set of in a such that x 1 x we prove that the commensurator of a x in a is equal to the normalizer of a x in a and that this group is generated by a x and the quasi centralizer of a x in a moreover if a x is not of type d l l 4 and l even then this group is generated by a x and the centralizer of a x in a

thick metric spaces relative hyperbolicity and quasi isometric rigidity we study the geometry of non relatively hyperbolic groups generalizing a result of schwartz any quasi isometric image of a non relatively hyperbolic space in a relatively hyperbolic space is contained in a bounded neighborhood of a single peripheral subgroup this implies that a group being relatively hyperbolic with non relatively hyperbolic peripheral subgroups is a quasi isometry invariant as an application artin groups are relatively hyperbolic if and only if freely decomposable we also introduce a new quasi isometry invariant of metric spaces called metrically thick which is sufficient for a metric space to be non hyperbolic relative to any non trivial collection of subsets thick finitely generated groups include mapping class groups of most surfaces outer automorphism groups of most free groups certain artin groups and others non uniform lattices in higher rank semisimple lie groups are thick and hence non relatively hyperbolic in contrast with rank one which provided the motivating examples of relatively hyperbolic groups mapping class groups are the first examples of non relatively hyperbolic groups having cut points in any asymptotic cone resolving several questions of drutu and sapir about the structure of relatively hyperbolic groups outside of group theory teichmuller spaces for surfaces of sufficiently large complexity are thick with respect to the weil peterson metric in contrast with brock farb s hyperbolicity result in low complexity

on the density of geometrically finite kleinian groups the density conjecture of bers sullivan and thurston predicts that each complete hyperbolic 3 manifold m with finitely generated fundamental group is an algebraic limit of geometrically finite hyperbolic 3 manifolds we prove that the conjecture obtains for each complete hyperbolic 3manifold with no cusps and incompressible ends

dimension and rank for mapping class groups we study the large scale geometry of the mapping class group mcg s our main result is that for any asymptotic cone of mcg s the maximal dimension of locally compact subsets coincides with the maximal rank of free abelian subgroups of mcg s an application is a proof of brock farb s rank conjecture which asserts that mcg s has quasi flats of dimension n if and only if it has a rank n free abelian subgroup hamenstadt has also given a proof of this conjecture using different methods we also compute the maximum dimension of quasi flats in teichmuller space with the weil petersson metric

related aspects of positivity in riemannian geometry pour une variete de riemann on etudie le lien entre le point 0 du l 2 spectre le plus proche de zero pour l extension de friedrich d un operateur semi defini symetrique et le taux exponentiel de transition d un operateur de markoff conservant la positivite

groups acting on tree graded spaces and splittings of relatively hyperbolic groups abstract tree graded spaces are generalizations of r trees they appear as asymptotic cones of groups when the cones have cut points since many questions about endomorphisms and automorphisms of groups solving equations over groups studying embeddings of a group into another group etc lead to actions of groups on the asymptotic cones it is natural to consider actions of groups on tree graded spaces we develop a theory of such actions which generalizes the well known theory of groups acting on r trees as applications of our theory we describe in particular relatively hyperbolic groups with infinite groups of outer automorphisms and co hopfian relatively hyperbolic groups

the hyperbolicity of the sphere complex via surgery paths handel and mosher have proved that the free splitting complex fs for the free group is gromov hyperbolic this is a deep and much sought after result since it establishes fs as a good analogue of the curve complex for surfaces we give a shorter alternative proof of this theorem using surgery paths in hatcher s sphere complex another model for the free splitting complex instead of handel and mosher s fold paths as a byproduct we get that surgery paths are unparameterized quasi geodesics in the sphere complex

hausdorff dimension and conformal dynamics i strong convergence of kleinian groups this paper investigates the behavior of the hausdorff dimensions of the limit sets n and of a sequence of kleinian groups n where m h is geometrically finite we show if n strongly then a mn h 3 n is geometrically finite for all n 0 b n in the hausdorff topology and c h dim n h dim if h dim 1 on the other hand we give examples showing the dimension can vary discontinuously under strong limits when h dim 1 continuity can be recovered by requiring that accidental parabolics converge radially similar results hold for higher dimensional manifolds applications are given to quasifuchsian groups and their limits

the classification of punctured torus groups thurston s ending lamination conjecture proposes that a flnitely generated kleinian group is uniquely determined up to isometry by the topology of its quotient and a list of invariants that describe the asymptotic geometry of its ends we present a proof of this conjecture for punctured torus groups these are free two generator kleinian groups with parabolic commutator which should be thought of as representations of the fundamental group of a punctured torus as a consequence we verify the conjectural topological description of the deformation space of punctured torus groups including bers conjecture that the quasi fuchsian groups are dense in this space and prove a rigidity theorem two punctured torus groups are quasi conformally conjugate if and only if they are topologically conjugate

on the braid index of alternating links we show that at least for an alternating fibered link or 2 bridge link l there is an exact formula which expresses the braid index b l of l as a function of the 2 variable generalization pl i m of the jones polynomial

hyperbolicity of the genus two hatcher thurston complex for the genus 1 surface with n punctures f 1 n we show that the hatcher thurston complex fancyscript ht f 1 n is hyperbolic for the genus 2 closed surface f 2 we show that the hatcher thurston complex fancyscript ht f 2 is strongly relatively hyperbolic with respect to a family of its subspaces x where ranges over all separating curves of f 2 and each x is isometric to the product of two farey graphs

quasi actions on trees i bounded valence given a bounded valence bushy tree t we prove that any cobounded quasi action of a group g on t is quasiconjugate to an action of g on another bounded valence bushy tree t this theorem has many applications quasi isometric rigidity for fundamental groups of finite bushy graphs of coarse pd n groups for each fixed n a generalization to actions on cantor sets of sullivan s theorem about uniformly quasiconformal actions on the 2 sphere and a characterization of locally compact topological groups which contain a virtually free group as a cocompact lattice finally we give the first examples of two finitely generated groups which are quasi isometric and yet which cannot act on the same proper geodesic metric space properly discontinuously and cocompactly by isometries

spectrally similar incommensurable 3 manifolds reid has asked whether hyperbolic manifolds with the same geodesic length spectrum must be commensurable building toward a negative answer to this question we construct examples of hyperbolic 3 manifolds that share an arbitrarily large portion of the length spectrum but are not commensurable more precisely for every n 0 we construct a pair of incommensurable hyperbolic 3 manifolds nn and nn whose volume is approximately n and whose length spectra agree up to length n both nn and nn are built by gluing two standard submanifolds along a complicated pseudo anosov map ensuring that these manifolds have a very thick collar about an essential surface the two gluing maps differ by a hyper elliptic involution along this surface our proof also involves a new commensurability criterion based on pairs of pants

quasi isometries preserve the geometric decomposition of haken manifolds we prove quasi isometry invariance of the canonical decomposition for fundamental groups of haken 3 manifolds with zero euler characteristic we show that groups quasi isometric to haken manifold groups with nontrivial canonical decomposition are finite extensions of haken orbifold groups as a by product we describe all 2 dimensional quasi flats in the universal covers of non geometric haken manifolds

convexity of the renormalized volume of hyperbolic 3 manifolds the hessian of the renormalized volume of geometrically finite hyperbolic 3 manifolds without rank 1 cusps computed at the hyperbolic metric g rm geod with totally geodesic boundary of the convex core is shown to be a strictly positive bilinear form on the tangent space to teichm uller space the metric g rm geod is known from results of bonahon and storm to be an absolute minimum for the volume of the convex core we deduce the strict convexity of the functional volume of the convex core at its minimum point