694 research outputs found
Random length-spectrum rigidity for free groups
We say that a subset is \emph{spectrally rigid} if whenever
are points of the (unprojectivized) Outer space such that
for every then in \cvn. It is
well-known that itself is spectrally rigid; it also follows from the
result of Smillie and Vogtmann that there does not exist a finite spectrally
rigid subset of . We prove that if is a free basis of (where
) then almost every trajectory of a non-backtracking simple random walk
on with respect to is a spectrally rigid subset of .Comment: 12 pages, no figures; to appear in Proceedings of the American
Mathematical Society; updated ref to the Duchin-Leininger-Rafi pape
-trees and laminations for free groups II: The dual lamination of an -tree
This is the second part of a series of three articles which introduce
laminations for free groups (see math.GR/0609416 for the first part). Several
definition of the dual lamination of a very small action of a free group on an
-tree are given and proved to be equivalent.Comment: corrections of typos and minor updat
Conformal dimension and random groups
We give a lower and an upper bound for the conformal dimension of the
boundaries of certain small cancellation groups. We apply these bounds to the
few relator and density models for random groups. This gives generic bounds of
the following form, where is the relator length, going to infinity.
(a) 1 + 1/C < \Cdim(\bdry G) < C l / \log(l), for the few relator model,
and
(b) 1 + l / (C\log(l)) < \Cdim(\bdry G) < C l, for the density model, at
densities .
In particular, for the density model at densities , as the relator
length goes to infinity, the random groups will pass through infinitely
many different quasi-isometry classes.Comment: 32 pages, 4 figures. v2: Final version. Main result improved to
density < 1/16. Many minor improvements. To appear in GAF
Genericity in Topological Dynamics
We study genericity of dynamical properties in the space of homeomorphisms of
the Cantor set and in the space of subshifts of a suitably large shift space.
These rather different settings are related by a Glasner-King type
correspondence: genericity in one is equivalent to genericity in the other.
By applying symbolic techniques in the shift-space model we derive new
results about genericity of dynamical properties for transitive and totally
transitive homeomorphisms of the Cantor set. We show that the isomorphism class
of the universal odometer is generic in the space of transitive systems. On the
other hand, the space of totally transitive systems displays much more varied
dynamics. In particular, we show that in this space the isomorphism class of
every Cantor system without periodic points is dense, and the following
properties are generic: minimality, zero entropy, disjointness from a fixed
totally transitive system, weak mixing, strong mixing, and minimal self
joinings. The last two stand in striking contrast to the situation in the
measure-preserving category. We also prove a correspondence between genericity
of dynamical properties in the measure-preserving category and genericity of
systems supporting an invariant measure with the same property.Comment: 48 pages, to appear in Ergodic Theory Dynamical Systems. v2: revised
exposition, added proof that the universal odometer is generic among
transitive Cantor homeomorphism
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