880 research outputs found
Weak hyperbolicity of cube complexes and quasi-arboreal groups
We examine a graph encoding the intersection of hyperplane carriers
in a CAT(0) cube complex . The main result is that is
quasi-isometric to a tree. This implies that a group acting properly and
cocompactly on 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 .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 , providing a class of groups
for which the simplicial boundary of a -cocompact cube complex depends only
on . We also use this result to show that the cocompactly cubulated
crystallographic groups in dimension 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 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 in a separable Hilbert space there exists a
(non-unique) Hilbert-Schmidt operator such that the perturbed
operator 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 in
and fixing an extension . We show that for a wide class of
symmetric operators the absolutely continuous parts of extensions and are unitarily equivalent provided that their
resolvent difference is a compact operator. Namely, we show that this is true
whenever the Weyl function of a pair admits bounded
limits M(t) := \wlim_{y\to+0}M(t+iy) for a.e. . 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 ,
where and are contractions on Hilbert space with trace class
difference, i.e., and is a function analytic in
the unit disk . It is well known that if is an operator Lipschitz
function analytic in , then . The main
result of the note says that there exists a function (a
spectral shift function) on the unit circle of class
such that the following trace formula holds:
, whenever and are
contractions with and is an operator Lipschitz
function analytic in .Comment: 6 page
- …