880 research outputs found

### Weak hyperbolicity of cube complexes and quasi-arboreal groups

We examine a graph $\Gamma$ encoding the intersection of hyperplane carriers
in a CAT(0) cube complex $\widetilde X$. The main result is that $\Gamma$ is
quasi-isometric to a tree. This implies that a group $G$ acting properly and
cocompactly on $\widetilde X$ 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 $\Gamma$.Comment: Corrections in Sections 2 and 4. Simplification in Section

### Cocompactly cubulated crystallographic groups

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 $S^{n-1}$, providing a class of groups $G$
for which the simplicial boundary of a $G$-cocompact cube complex depends only
on $G$. We also use this result to show that the cocompactly cubulated
crystallographic groups in dimension $n$ 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

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 $G$ 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

The classical Weyl-von Neumann theorem states that for any self-adjoint
operator $A$ in a separable Hilbert space $\mathfrak H$ there exists a
(non-unique) Hilbert-Schmidt operator $C = C^*$ such that the perturbed
operator $A+C$ 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 $A$ in $\mathfrak H$
and fixing an extension $A_0 = A_0^*$. We show that for a wide class of
symmetric operators the absolutely continuous parts of extensions $\widetilde A
= {\widetilde A}^*$ and $A_0$ are unitarily equivalent provided that their
resolvent difference is a compact operator. Namely, we show that this is true
whenever the Weyl function $M(\cdot)$ of a pair $\{A,A_0\}$ admits bounded
limits M(t) := \wlim_{y\to+0}M(t+iy) for a.e. $t \in \mathbb{R}$. 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)

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

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

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