241 research outputs found
Convex cocompactness and stability in mapping class groups
We introduce a strong notion of quasiconvexity in finitely generated groups,
which we call stability. Stability agrees with quasiconvexity in hyperbolic
groups and is preserved under quasi-isometry for finitely generated groups. We
show that the stable subgroups of mapping class groups are precisely the convex
cocompact subgroups. This generalizes a well-known result of Behrstock and is
related to questions asked by Farb-Mosher and Farb.Comment: 15 pages, 1 figur
The augmented marking complex of a surface
We build an augmentation of the Masur-Minsky marking complex by
Groves-Manning combinatorial horoballs to obtain a graph we call the augmented
marking complex, . Adapting work of Masur-Minsky, we prove
that is quasiisometric to Teichm\"uller space with the
Teichm\"uller metric. A similar construction was independently discovered by
Eskin-Masur-Rafi. We also completely integrate the Masur-Minsky hierarchy
machinery to to build flexible families of uniform
quasigeodesics in Teichm\"uller space. As an application, we give a new proof
of Rafi's distance formula for the Teichm\"uller metric.Comment: 30 pages; significantly rewritten to strengthen main construction
Anabelian Intersection Theory I: The Conjecture of Bogomolov-Pop and Applications
We finish the proof of the conjecture of F. Bogomolov and F. Pop: Let
and be fields finitely-generated and of transcendence degree
over and , respectively, where is either
or , and is algebraically
closed. We denote by and their respective absolute Galois
groups. Then the canonical map \varphi_{F_{1}, F_{2}}: \Isom^i(F_1,
F_2)\rightarrow \Isom^{\Out}_{\cont}(G_{F_2}, G_{F_1}) from the isomorphisms,
up to Frobenius twists, of the inseparable closures of and to
continuous outer isomorphisms of their Galois groups is a bijection. Thus,
function fields of varieties of dimension over algebraic closures of
prime fields are anabelian. We apply this to give a necessary and sufficient
condition for an element of the Grothendieck-Teichm\"uller group to be an
element of the absolute Galois group of .Comment: 30 pages, comments welcome
The extended hypergeometric class of L\'evy processes
With a view to computing fluctuation identities related to stable processes,
we review and extend the class of hypergeometric L\'evy processes explored in
Kuznetsov and Pardo (arXiv:1012.0817). We give the Wiener-Hopf factorisation of
a process in the extended class, and characterise its exponential functional.
Finally, we give three concrete examples arising from transformations of stable
processes.Comment: 22 page
Source-Channel Secrecy with Causal Disclosure
Imperfect secrecy in communication systems is investigated. Instead of using
equivocation as a measure of secrecy, the distortion that an eavesdropper
incurs in producing an estimate of the source sequence is examined. The
communication system consists of a source and a broadcast (wiretap) channel,
and lossless reproduction of the source sequence at the legitimate receiver is
required. A key aspect of this model is that the eavesdropper's actions are
allowed to depend on the past behavior of the system. Achievability results are
obtained by studying the performance of source and channel coding operations
separately, and then linking them together digitally. Although the problem
addressed here has been solved when the secrecy resource is shared secret key,
it is found that substituting secret key for a wiretap channel brings new
insights and challenges: the notion of weak secrecy provides just as much
distortion at the eavesdropper as strong secrecy, and revealing public messages
freely is detrimental.Comment: Allerton 2012, 6 pages. Updated version includes acknowledgement
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