880 research outputs found

    Weak hyperbolicity of cube complexes and quasi-arboreal groups

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    We examine a graph Γ\Gamma encoding the intersection of hyperplane carriers in a CAT(0) cube complex X~\widetilde X. The main result is that Γ\Gamma is quasi-isometric to a tree. This implies that a group GG acting properly and cocompactly on X~\widetilde X is weakly hyperbolic relative to the hyperplane stabilizers. Using disc diagram techniques and Wright's recent result on the aymptotic dimension of CAT(0) cube complexes, we give a generalization of a theorem of Bell and Dranishnikov on the finite asymptotic dimension of graphs of asymptotically finite-dimensional groups. More precisely, we prove asymptotic finite-dimensionality for finitely-generated groups acting on finite-dimensional cube complexes with 0-cube stabilizers of uniformly bounded asymptotic dimension. Finally, we apply contact graph techniques to prove a cubical version of the flat plane theorem stated in terms of complete bipartite subgraphs of Γ\Gamma.Comment: Corrections in Sections 2 and 4. Simplification in Section

    Cocompactly cubulated crystallographic groups

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    We prove that the simplicial boundary of a CAT(0) cube complex admitting a proper, cocompact action by a virtually \integers^n group is isomorphic to the hyperoctahedral triangulation of Sn1S^{n-1}, providing a class of groups GG for which the simplicial boundary of a GG-cocompact cube complex depends only on GG. We also use this result to show that the cocompactly cubulated crystallographic groups in dimension nn are precisely those that are \emph{hyperoctahedral}. We apply this result to answer a question of Wise on cocompactly cubulating virtually free abelian groups.Comment: Several correction

    On hierarchical hyperbolicity of cubical groups

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    Let X be a proper CAT(0) cube complex admitting a proper cocompact action by a group G. We give three conditions on the action, any one of which ensures that X has a factor system in the sense of [BHS14]. We also prove that one of these conditions is necessary. This combines with results of Behrstock--Hagen--Sisto to show that GG is a hierarchically hyperbolic group; this partially answers questions raised by those authors. Under any of these conditions, our results also affirm a conjecture of BehrstockHagen on boundaries of cube complexes, which implies that X cannot contain a convex staircase. The conditions on the action are all strictly weaker than virtual cospecialness, and we are not aware of a cocompactly cubulated group that does not satisfy at least one of the conditions.Comment: Minor changes in response to referee report. Streamlined the proof of Lemma 5.2, and added an examples of non-rotational action

    On the unitary equivalence of absolutely continuous parts of self-adjoint extensions

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    The classical Weyl-von Neumann theorem states that for any self-adjoint operator AA in a separable Hilbert space H\mathfrak H there exists a (non-unique) Hilbert-Schmidt operator C=CC = C^* such that the perturbed operator A+CA+C has purely point spectrum. We are interesting whether this result remains valid for non-additive perturbations by considering self-adjoint extensions of a given densely defined symmetric operator AA in H\mathfrak H and fixing an extension A0=A0A_0 = A_0^*. We show that for a wide class of symmetric operators the absolutely continuous parts of extensions A~=A~\widetilde A = {\widetilde A}^* and A0A_0 are unitarily equivalent provided that their resolvent difference is a compact operator. Namely, we show that this is true whenever the Weyl function M()M(\cdot) of a pair {A,A0}\{A,A_0\} admits bounded limits M(t) := \wlim_{y\to+0}M(t+iy) for a.e. tRt \in \mathbb{R}. This result is applied to direct sums of symmetric operators and Sturm-Liouville operators with operator potentials

    Bilateral Differentiation of Color and Morphology in the Larval and Pupal Stages of \u3ci\u3ePapilio Glaucus\u3c/i\u3e (Lepidoptera: Papilionidae)

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    A sharply delineated, bilateral differentiation of color patterns and morphology were observed in a final (5th) instar larva of a subspecies backcross of a female Papilio glaucus glaucus with a hybrid male (P. g. glaucus x P. g. canadensis). Color and morphological differences were detectable in the pupa as well. In addition, a bilateral size difference was evident in both the pupa and the resulting adult butterfly. Such observations within a single living individual attest to the bilateral independence (also evident in perfect gynandromorphs) and general flexibility of the developmental control in this species of Lepidoptera

    A trace formula for functions of contractions and analytic operator Lipschitz functions

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    In this note we study the problem of evaluating the trace of f(T)f(R)f(T)-f(R), where TT and RR are contractions on Hilbert space with trace class difference, i.e., TRS1T-R\in\boldsymbol{S}_1 and ff is a function analytic in the unit disk D{\Bbb D}. It is well known that if ff is an operator Lipschitz function analytic in D{\Bbb D}, then f(T)f(R)S1f(T)-f(R)\in\boldsymbol{S}_1. The main result of the note says that there exists a function ξ\boldsymbol{\xi} (a spectral shift function) on the unit circle T{\Bbb T} of class L1(T)L^1({\Bbb T}) such that the following trace formula holds: trace(f(T)f(R))=Tf(ζ)ξ(ζ)dζ\operatorname{trace}(f(T)-f(R))=\int_{\Bbb T} f'(\zeta)\boldsymbol{\xi}(\zeta)\,d\zeta, whenever TT and RR are contractions with TRS1T-R\in\boldsymbol{S}_1 and ff is an operator Lipschitz function analytic in D{\Bbb D}.Comment: 6 page