2,143 research outputs found
Acylindrical hyperbolicity of cubical small-cancellation groups
We provide an analogue of Strebel's classification of geodesic triangles in
classical groups for groups given by Wise's cubical presentations
satisfying sufficiently strong metric cubical small cancellation conditions.
Using our classification, we prove that, except in specific degenerate cases,
such groups are acylindrically hyperbolic.Comment: Added figures. Exposition improved in Section 3,
correction/simplification in Section 5, background added and citations
updated in Section
Geometry of dynamics, Lyapunov exponents and phase transitions
The Hamiltonian dynamics of classical planar Heisenberg model is numerically
investigated in two and three dimensions. By considering the dynamics as a
geodesic flow on a suitable Riemannian manifold, it is possible to analytically
estimate the largest Lyapunov exponent in terms of some curvature fluctuations.
The agreement between numerical and analytical values for Lyapunov exponents is
very good in a wide range of temperatures. Moreover, in the three dimensional
case, in correspondence with the second order phase transition, the curvature
fluctuations exibit a singular behaviour which is reproduced in an abstract
geometric model suggesting that the phase transition might correspond to a
change in the topology of the manifold whose geodesics are the motions of the
system.Comment: REVTeX, 10 pages, 5 PostScript figures, published versio
Systoles and kissing numbers of finite area hyperbolic surfaces
We study the number and the length of systoles on complete finite area
orientable hyperbolic surfaces. In particular, we prove upper bounds on the
number of systoles that a surface can have (the so-called kissing number for
hyperbolic surfaces). Our main result is a bound which only depends on the
topology of the surface and which grows subquadratically in the genus.Comment: A minor mistake and a computation fixed, small changes in the
exposition. 23 pages, 13 figure
Local geometry of random geodesics on negatively curved surfaces
It is shown that the tessellation of a compact, negatively curved surface
induced by a typical long geodesic segment, when properly scaled, looks locally
like a Poisson line process. This implies that the global statistics of the
tessellation -- for instance, the fraction of triangles -- approach those of
the limiting Poisson line process.Comment: This version extends the results of the previous version to surfaces
with possibly variable negative curvatur
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