204 research outputs found

    A Note on Complex-Hyperbolic Kleinian Groups

    Get PDF
    Let Γ be a discrete group of isometries acting on the complex hyperbolic n-space HCn. In this note, we prove that if Γ is convex-cocompact, torsion-free, and the critical exponent δ(Γ) is strictly lesser than 2, then the complex manifold HCn/Γ is Stein. We also discuss several related conjectures

    Subset currents on free groups

    Full text link
    We introduce and study the space of \emph{subset currents} on the free group FNF_N. A subset current on FNF_N is a positive FNF_N-invariant locally finite Borel measure on the space CN\mathfrak C_N of all closed subsets of FN\partial F_N consisting of at least two points. While ordinary geodesic currents generalize conjugacy classes of nontrivial group elements, a subset current is a measure-theoretic generalization of the conjugacy class of a nontrivial finitely generated subgroup in FNF_N, and, more generally, in a word-hyperbolic group. The concept of a subset current is related to the notion of an "invariant random subgroup" with respect to some conjugacy-invariant probability measure on the space of closed subgroups of a topological group. If we fix a free basis AA of FNF_N, a subset current may also be viewed as an FNF_N-invariant measure on a "branching" analog of the geodesic flow space for FNF_N, whose elements are infinite subtrees (rather than just geodesic lines) of the Cayley graph of FNF_N with respect to AA.Comment: updated version; to appear in Geometriae Dedicat

    A note on Selberg's lemma and negatively curved Hadamard manifolds

    Get PDF
    Answering a question by Margulis we prove that the conclusion of Selberg's Lemma fails for discrete isometry groups of negatively curved Hadamard manifolds

    Parabolic groups acting on one-dimensional compact spaces

    Full text link
    Given a class of compact spaces, we ask which groups can be maximal parabolic subgroups of a relatively hyperbolic group whose boundary is in the class. We investigate the class of 1-dimensional connected boundaries. We get that any non-torsion infinite f.g. group is a maximal parabolic subgroup of some relatively hyperbolic group with connected one-dimensional boundary without global cut point. For boundaries homeomorphic to a Sierpinski carpet or a 2-sphere, the only maximal parabolic subgroups allowed are virtual surface groups (hyperbolic, or virtually Z+Z\mathbb{Z} + \mathbb{Z}).Comment: 10 pages. Added a precision on local connectedness for Lemma 2.3, thanks to B. Bowditc

    Quantum geometry from phase space reduction

    Full text link
    In this work we give an explicit isomorphism between the usual spin network basis and the direct quantization of the reduced phase space of tetrahedra. The main outcome is a formula that describes the space of SU(2) invariant states by an integral over coherent states satisfying the closure constraint exactly, or equivalently, as an integral over the space of classical tetrahedra. This provides an explicit realization of theorems by Guillemin--Sternberg and Hall that describe the commutation of quantization and reduction. In the final part of the paper, we use our result to express the FK spin foam model as an integral over classical tetrahedra and the asymptotics of the vertex amplitude is determined.Comment: 33 pages, 1 figur

    Stabilizers of R\mathbb R-trees with free isometric actions of FNF_N

    Full text link
    We prove that if TT is an R\mathbb R-tree with a minimal free isometric action of FNF_N, then the Out(FN)Out(F_N)-stabilizer of the projective class [T][T] is virtually cyclic. For the special case where T=T+(ϕ)T=T_+(\phi) is the forward limit tree of an atoroidal iwip element ϕOut(FN)\phi\in Out(F_N) this is a consequence of the results of Bestvina, Feighn and Handel, via very different methods. We also derive a new proof of the Tits alternative for subgroups of Out(FN)Out(F_N) containing an iwip (not necessarily atoroidal): we prove that every such subgroup GOut(FN)G\le Out(F_N) is either virtually cyclic or contains a free subgroup of rank two. The general case of the Tits alternative for subgroups of Out(FN)Out(F_N) is due to Bestvina, Feighn and Handel.Comment: corrected the proof of Proposition 4.1, plus several minor fixes and updates; to appear in Journal of Group Theor

    Intersection form, laminations and currents on free groups

    Get PDF
    Let FNF_N be a free group of rank N2N\ge 2, let μ\mu be a geodesic current on FNF_N and let TT be an R\mathbb R-tree with a very small isometric action of FNF_N. We prove that the geometric intersection number is equal to zero if and only if the support of μ\mu is contained in the dual algebraic lamination L2(T)L^2(T) of TT. Applying this result, we obtain a generalization of a theorem of Francaviglia regarding length spectrum compactness for currents with full support. As another application, we define the notion of a \emph{filling} element in FNF_N and prove that filling elements are "nearly generic" in FNF_N. We also apply our results to the notion of \emph{bounded translation equivalence} in free groups.Comment: revised version, to appear in GAF

    From twistors to twisted geometries

    Full text link
    In a previous paper we showed that the phase space of loop quantum gravity on a fixed graph can be parametrized in terms of twisted geometries, quantities describing the intrinsic and extrinsic discrete geometry of a cellular decomposition dual to the graph. Here we unravel the origin of the phase space from a geometric interpretation of twistors.Comment: 9 page

    Discreteness of the volume of space from Bohr-Sommerfeld quantization

    Full text link
    A major challenge for any theory of quantum gravity is to quantize general relativity while retaining some part of its geometrical character. We present new evidence for the idea that this can be achieved by directly quantizing space itself. We compute the Bohr-Sommerfeld volume spectrum of a tetrahedron and show that it reproduces the quantization of a grain of space found in loop gravity.Comment: 4 pages, 4 figures; v2, to appear in PR

    A Bi-Hamiltonian Structure for the Integrable, Discrete Non-Linear Schrodinger System

    Full text link
    This paper shows that the Ablowitz-Ladik hierarchy of equations (a well-known integrable discretization of the Non-linear Schrodinger system) can be explicitly viewed as a hierarchy of commuting flows which: (a) are Hamiltonian with respect to both a standard, local Poisson operator J and a new non-local, skew, almost Poisson operator K, on the appropriate space; (b) can be recursively generated from a recursion operator R (obtained by composing K and the inverse of J.) In addition, the proof of these facts relies upon two new pivotal resolvent identities which suggest a general method for uncovering bi-Hamiltonian structures for other families of discrete, integrable equations.Comment: 33 page
    corecore