45,171 research outputs found
The isomorphism problem for all hyperbolic groups
We give a solution to Dehn's isomorphism problem for the class of all
hyperbolic groups, possibly with torsion. We also prove a relative version for
groups with peripheral structures. As a corollary, we give a uniform solution
to Whitehead's problem asking whether two tuples of elements of a hyperbolic
group are in the same orbit under the action of \Aut(G). We also get an
algorithm computing a generating set of the group of automorphisms of a
hyperbolic group preserving a peripheral structure.Comment: 71 pages, 4 figure
On metrics of curvature 1 with four conic singularities on tori and on the sphere
We discuss conformal metrics of curvature 1 on tori and on the sphere, with
four conic singularities whose angles are multiples of pi/2. Besides some
general results we study in detail the family of such symmetric metrics on the
sphere, with angles (pi/2,3pi/2,pi/2,3pi/2).Comment: 25 pages, 5 figure
Two-Variable Logic with Two Order Relations
It is shown that the finite satisfiability problem for two-variable logic
over structures with one total preorder relation, its induced successor
relation, one linear order relation and some further unary relations is
EXPSPACE-complete. Actually, EXPSPACE-completeness already holds for structures
that do not include the induced successor relation. As a special case, the
EXPSPACE upper bound applies to two-variable logic over structures with two
linear orders. A further consequence is that satisfiability of two-variable
logic over data words with a linear order on positions and a linear order and
successor relation on the data is decidable in EXPSPACE. As a complementing
result, it is shown that over structures with two total preorder relations as
well as over structures with one total preorder and two linear order relations,
the finite satisfiability problem for two-variable logic is undecidable
Interior design of a two-dimensional semiclassical black hole: Quantum transition across the singularity
We study the internal structure of a two-dimensional dilatonic evaporating
black hole, based on the CGHS model. At the semiclassical level, a (weak)
spacelike singularity was previously found to develop inside the black hole. We
employ here a simplified quantum formulation of spacetime dynamics in the
neighborhood of this singularity, using a minisuperspace-like approach. Quantum
evolution is found to be regular and well-defined at the semiclassical
singularity. A well-localized initial wave-packet propagating towards the
singularity bounces off the latter and retains its well-localized form. Our
simplified quantum treatment thus suggests that spacetime may extend
semiclassically beyond the singularity, and also signifies the specific
extension.Comment: Accepted to Phys. Rev.
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