10,931 research outputs found
On the acylindrical hyperbolicity of the tame automorphism group of
We introduce the notion of \"uber-contracting element, a strengthening of the
notion of strongly contracting element which yields a particularly tractable
criterion to show the acylindrical hyperbolicity, and thus a strong form of
non-simplicity, of groups acting on non locally compact spaces of arbitrary
dimension. We also give a simple local criterion to construct
\"uber-contracting elements for groups acting on complexes with unbounded links
of vertices.
As an application, we show the acylindrical hyperbolicity of the tame
automorphism group of , a subgroup of the
-dimensional Cremona group, through its action on a CAT(0) square complex
recently introduced by Bisi-Furter-Lamy.Comment: 16 pages, 1 figur
On the cubical geometry of Higman's group
We investigate the cocompact action of Higman's group on a CAT(0) square
complex associated to its standard presentation. We show that this action is in
a sense intrinsic, which allows for the use of geometric techniques to study
the endomorphisms of the group, and show striking similarities with mapping
class groups of hyperbolic surfaces, outer automorphism groups of free groups
and linear groups over the integers. We compute explicitly the automorphism
group and outer automorphism group of Higman's group, and show that the group
is both hopfian and co-hopfian. We actually prove a stronger rigidity result
about the endomorphisms of Higman's group: Every non-trivial morphism from the
group to itself is an automorphism. We also study the geometry of the action
and prove a surprising result: Although the CAT(0) square complex acted upon
contains uncountably many flats, the Higman group does not contain subgroups
isomorphic to Z^2. Finally, we show that this action possesses features
reminiscent of negative curvature, which we use to prove a refined version of
the Tits alternative for Higman's group.Comment: Accepted versio
Optimal synchronization deep in the quantum regime: resource and fundamental limit
We develop an analytical framework to study the synchronization of a quantum
self-sustained oscillator to an external signal. Our unified description allows
us to identify the resource on which quantum synchronization relies, and to
compare quantitatively the synchronization behavior of different limit cycles
and signals. We focus on the most elementary quantum system that is able to
host a self-sustained oscillation, namely a single spin 1. Despite the spin
having no classical analogue, we first show that it can realize the van der Pol
limit cycle deep in the quantum regime, which allows us to provide an
analytical understanding to recently reported numerical results. Moving on to
the equatorial limit cycle, we then reveal the existence of an
interference-based quantum synchronization blockade and extend the classical
Arnold tongue to a snake-like split tongue. Finally, we derive the maximum
synchronization that can be achieved in the spin-1 system, and construct a
limit cycle that reaches this fundamental limit asymptotically.Comment: 15 pages, 9 figures, equivalent to published versio
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