the topology of deformation spaces of kleinian groups let m be a compact hyperbolizable 3 manifold with nonempty incompressible boundary and let ah 1 m denote the space of conjugacy classes of discrete faithful representations of 1 m i nto psl 2 c the components of the interior mp 1 m of ah 1 m as a subset of the appropriate representation variety are enumerated by the spacea m of marked homeomorphism types of oriented compact irreducible 3 manifolds homotopy equivalent to m in this paper we give a topological enumeration of the components of the closure of mp 1 m and hence a conjectural topological enumeration of the components of ah 1 m we do so by characterizing exactly which changes of marked homeomorphism type can occur in the algebraic limit of a sequence of isomorphic freely indecomposable kleinian groups we use this enumeration to exhibit manifolds m for which ah 1 m has inflnitely many components in this paper we begin a study of the global topology of deformation spaces of kleinian groups the basic object of study is the space ah 1 m of marked hyperbolic 3 manifolds homotopy equivalent to a flxed compact 3manifold m the interior mp 1 m of ah 1 m is very well understood due to work of ahlfors bers kra marden maskit sullivan and thurston in particular the components ofmp 1 m are enumerated by topological data namely the set a m of marked compact oriented irreducible 3 manifolds homotopy equivalent to m while each component is parametrized by analytic data coming from the conformal boundaries of the hyperbolic 3 manifolds thurston s ending lamination conjecture provides a conjectural classiflcation for elements of ah 1 m by data which are partially topologi

on the equality of integrals considering a finite space x several necessary conditions and one rather general sufficient condition describing when the choquet integral coincides with the pan integral with respect to the standard arithmetic operations are shown these conditions are characterized by using the minimal atoms of monotone measure under the introduced constraints the calculation of these coinciding two integrals is also given

a sufficient condition of equivalence of the choquet and the pan integral abstract in this note we continue to investigate the relationship between the choquet integral and the pan integral on infinite space we will show that the m property of monotone measures is a sufficient condition that the choquet integral coincides with the pan integral in this discussion the spaces are not restricted to be finite thus the previous results obtained in finite space are further generalized and developed a characteristic of the m property of monotone measures is also presented

singular surfaces mod 2 homology and hyperbolic volume i if m is a simple closed orientable 3 manifold such that 1 m contains a genus g surface group and if h 1 m 2 has rank at least 4g 1 we show that m contains an embedded closed incompressible surface of genus at most g as an application we show that if m is a closed orientable hyperbolic 3 manifold of volume at most 3 08 then the rank of h 1 m 2 is at most 6

growth rates _ homology and volumes of hyperbolic 3 manifolds it is shown that if m is a closed orientable irreducible 3 manifold and n is a nonnegative integer and if h 1 m z p has rank n 2 for some prime p then every n generator subgroup of 1 m has infinite index in 1 m and is in fact contained in infinitely many finite index subgroups of 1 m this result is used to estimate the growth rates of the fundamental group of a 3 manifold in terms of the rank of the z p homology

generalized two mode harmonic oscillator dynamical group and squeezed states in this paper we generalize our previous results of the generalized one mode harmonic oscillator to the generalized two mode case systematic use is made of the so 3 2 dynamical group and we are able to write a general form for the exact time evolution operator in terms of squeezing operators of one and two modes a complete classification of the exact solutions is made and we derive them explicitly whenever possible the relevant results on algebraic decomposition coherent state generators and classification of the solutions are shown in tables a plethora of soluble hamiltonians already treated in the literature which appear to be particular cases of the general formalism presented herein are analysed as well as new cases which to the authors knowledge have not yet been considered

a note on the coincidence of decomposition integrals and superdecomposition integrals abstract it is known that the decomposition integral i h and the superdecomposition integral i h related to the system h of all finite chains from a where a is a algebra of the subsets of a nonvoid set x coincide with each other for each monotone measure and for each nonnegative measurable function and they are equal to the choquet integral in this note we show that the converse is also true that is for a complete system h of finite set systems from a if i h i h holds for each monotone measure and for each nonnegative measurable function then the system h and the system of all finite chains from a are equivalent in some sense and the integrals involved are the choquet integral

on the equivalence of the choquet pan and concave integrals on finite spaces abstract in this paper we introduce the concept of maximal cluster of minimal atoms on monotone measure spaces and by means of this new concept we continue to investigate the relation between the choquet integral and the pan integral on finite spaces it is proved that the m property of a monotone measure is a sufficient condition that the choquet integral coincides with the pan integral based on the usual addition and multiplication thus combining our recent results we provide a necessary and sufficient condition that the choquet integral is equivalent to the pan integral on finite spaces meanwhile we also use the characteristics of minimal atoms of monotone measure to present another necessary and sufficient condition that these two kinds of integrals are equivalent on finite spaces the relationships among the choquet integral the pan integral and the concave integral are summarized

on continuous extensions of grafting maps the definition of the grafting operation for quasifuchsian groups is extended by bromberg preprint to all b groups in this paper we show that the extended grafting maps behave as continuous maps for every sequence which converges standardly to a boundary group although the maps are not continuous in general as a consequence of this result we extend goldman s grafting theorem for quasifuchsian groups to all boundary b groups

cusps are dense we show cusps are dense in bers boundary for teichmuller space the proof rests on an estimate for the algebraic effect of a unit quasiconformal deformation supported in the thin part of a hyperbolic riemann surface

approximation by maximal cusps in boundaries of deformation spaces of kleinian groups let m be a compact oriented irreducible atoroidal 3 manifold with nonempty boundary let cc0 m denote the space of convex cocompact kleinian groups uniformizing m we show that any kleinian group in the boundary of cc0 m whose limit set is the whole sphere can be approximated by maximal cusps density of maximal cusps on the boundary of schottky space is derived as a corollary we further show that maximal cusps are dense in the boundary of the quasiconformal deformation space of any geometrically finite hyperbolic 3 manifold with connected conformal boundary

the visual core of a hyperbolic 3 manifold in this note we introduce the notion of the visual core of a hyperbolic 3 manifold n bf h 3 gamma and explore some of its basic properties we investigate circumstances under which the visual core cal v n of a cover n bf h 3 gamma of n embeds in n via the usual covering map pi n rightarrow n we go on to show that if the algebraic limit of a sequence of isomorphic kleinian groups is a generalized web group then the visual core of the algebraic limit manifold embeds in the geometric limit manifold finally we discuss the relationship between the visual core and klein maskit combination along component subgroups

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

geometrical finiteness for hyperbolic groups abstract in this paper we give an account of the notion of geometrical finiteness as applied to discrete groups acting on hyperbolic space of any dimension we prove the equivalence of various definitions of geometrical finiteness and describe the geometry of fundamental domains we give a complete account of when dirichiet domains are finite sided

relationship between two types of superdecomposition integrals on finite spaces abstract this paper investigates the relationship between two types of superdecomposition integrals namely the convex integral and the pan integral from above on finite spaces to this end we introduce two new concepts related to monotone measures superadditivity with respect to singletons and minimal strictly subadditive set and discuss some of their properties in the case that the monotone measure is superadditive with respect to singletons we show that these two types of integrals are equivalent in other cases by means of the characteristics of minimal strictly subadditive sets we provide a set of necessary and sufficient conditions for which these two types of integrals coincide with each other

superdecomposition integrals this study introduces and discusses a new class of integrals based on superdecompositions of integrated functions including an analysis of their relationship with decomposition integrals which were introduced recently by even and lehrer the proposed superdecomposition integrals have several properties that are similar or dual with respect to decomposition integrals but they also have some significant differences the convex integral is obtained by considering all possible superdecompositions with no constraints on the applied sets which can be treated as the greatest convex homogeneous functional that is bounded from above by the measure we consider the relationship with the universal integral of klement et al is also discussed finally some possible generalizations are outlined

su 1 1 intelligent states we construct all the intelligent states of the non compact generators of su 1 1 for every positive discrete representation of this lie algebra and discuss some of the properties of these states

su 1 1 symmetry of multimode squeezed states we show that a class of multimode optical transformations that employ linear optics plus two mode squeezing can be expressed as su 1 1 operators these operations are relevant to state of the art continuous variable quantum information experiments including quantum state sharing quantum teleportation and multipartite entangled states using this su 1 1 description of these transformations we obtain a new basis for such transformations that lies in a useful representation of this group and lies outside the often used restriction to gaussian states we analyze this basis show its application to a class of transformations and discuss its extension to more general quantum optical networks

the classification of kleinian surface groups ii the ending lamination conjecture thurston s ending lamination conjecture states that a hyperbolic 3manifold n with nitely generated fundamental group is uniquely determined by its topological type and its end invariants in this paper we prove this conjecture for kleinian surface groups the general case when n has incompressible ends relative to its cusps follows readily the main ingredient is a uniformly bilipschitz model for the quotient of h 3 by a kleinian surface group

coherent state triplets and their inner products it is shown that if h is a hilbert space for a representation of a group g then there are triplets of spaces fh h fh in which fh is a space of coherent state or vector coherent state wave functions and fh is its dual relative to a conveniently defined measure it is shown also that there is a sequence of maps fh h fh which facilitates the construction of the corresponding inner products after completion if necessary the spaces fh h fh become isomorphic hilbert spaces it is shown that the inner product for h is often easier to evaluate in fh than in fh thus we obtain integral expressions for the inner products of coherent state and vector coherent state representations these expressions are equivalent to the algebraic expressions of k matrix theory but they are frequently more efficient to apply the construction is illustrated by many examples

relationship between the concave integrals and the pan integrals on finite spaces a r t i c l e i n f o a b s t r a c t this study discusses the relationship between the concave integrals and the pan integrals on finite spaces the minimal atom of a monotone measure is introduced and some properties are investigated by means of two important structure characteristics related to minimal atoms minimal atoms disjointness property and subadditivity for minimal atoms a necessary and sufficient condition is given that the concave integral coincides with the pan integral with respect to the standard arithmetic operations a nd on finite spaces following this result we have shown that these two integrals coincide if the underlying monotone measure is sub additive

choquet like integrals abstract integrals based on a pseudo addition and a pseudo multiplication with properties similar to those of the choquet and the sugeno integral respectively are introduced the structure of convenient pseudo arithmetical operations leading to a choquet like integral is studied

cores of hyperbolic 3 manifolds and limits of kleinian groups troels jorgensen conjectured that the algebraic and geometric limits of an algebraically convergent sequence of isomorphic kleinian groups agree if there are no new parabolics in the algebraic limit we prove that this conjecture holds in most cases in particular we show that it holds when the domain of discontinuity of the algebraic limit of such a sequence is non empty we further show with the same assumptions that the limit sets of the groups in the sequence converge to the limit set of the algebraic limit as a corollary we verify the conjecture for finitely generated kleinian groups which are not non trivial free products of surface groups and infinite cyclic groups these results are extensions of similar results for purely loxodromic groups thurston previously established these results in the case when the kleinian groups are freely indecomposable using different techniques from ours ohshika has proven versions of these results for purely loxodromic function groups

efficient sharing of a continuous variable quantum secret we propose an efficient scheme for sharing a continuous variable quantum secret using passive optical interferometry and squeezers this efficiency is achieved by showing that a maximum of two squeezers is required to extract the secret state and we obtain the cheapest configuration in terms of total squeezing cost squeezing is a cost for the dealer of the secret as well as for the receivers and we quantify limitations to the fidelity of the extracted secret state in terms of the squeezing employed by the dealer

the concave integral over large spaces this paper investigates the concave integral for capacities defined over large spaces we characterize when the integral with respect to capacity v can be represented as the infimum over all integrals with respect to additive measures that are greater than or equal to v we introduce the notion of loose extendability and study its relation to the concave integral a non additive version for the levi theorem and the fatou lemma are proven finally we provide several convergence theorems for capacities with large cores

on the equivalence of the choquet integral and the pan integrals from above abstract a monotone measure is said to have dual m property if for any a b there exists c with a c b such that c a and b c b c by using this concept we study the relationship of the choquet integral and the pan integral from above we prove that the choquet integral coincides with the pan integral from above if has dual m property when the underlying space is finite we prove that the dual m property is also necessary for the coincidence of these two integrals thus we provide a necessary and sufficient condition for the equivalence of the choquet integral and the pan integral from above on a finite space

on the m property of monotone measures and integrals on atoms abstract we further investigate m property an important structure characteristic of monotone measures some necessary and or sufficient conditions of m property are shown and some characteristics of m property are described it is shown that the restriction of a monotone measure to an atom of its own possesses m property on this atom by means of the m property we characterize the equivalence among the choquet integral the pan integral the concave integral and the shilkret integral on atoms of monotone measures and obtain some special results we also show a necessary and sufficient condition for the concave integral and the pan integral to coincide on atoms of monotone measures

quantum integrability in the multistate landau zener problem we analyze hamiltonians linear in the time variable for which the multistate landau zener lz problem is known to have an exact solution we show that they either belong to families of mutually commuting hamiltonians polynomial in time or reduce to the lz problem which is considered trivially integrable the former category includes the equal slope bow tie and generalized bow tie models for each of these models we explicitly construct the corresponding families of commuting matrices the equal slope model is a member of an integrable family that consists of the maximum possible number for a given matrix size of commuting matrices linear in time the bow tie model belongs to a previously unknown similarly maximal family of quadratic commuting matrices we thus conjecture that quantum integrability understood as the existence of nontrivial parameter dependent commuting partners is a necessary condition for the lz solvability descendants of the lz hamiltonian are e g general su 2 and hamiltonians time dependent linear chain linear nonlinear and double oscillators we explicitly obtain solutions to all these lz problems from the case