208 research outputs found
The braided Ptolemy-Thompson group is asynchronously combable
The braided Ptolemy-Thompson group is an extension of the Thompson
group by the full braid group on infinitely many strands. This
group is a simplified version of the acyclic extension considered by Greenberg
and Sergiescu, and can be viewed as a mapping class group of a certain infinite
planar surface. In a previous paper we showed that is finitely presented.
Our main result here is that (and ) is asynchronously combable. The
method of proof is inspired by Lee Mosher's proof of automaticity of mapping
class groups.Comment: 45
Non ambiguous structures on 3-manifolds and quantum symmetry defects
The state sums defining the quantum hyperbolic invariants (QHI) of hyperbolic
oriented cusped -manifolds can be split in a "symmetrization" factor and a
"reduced" state sum. We show that these factors are invariants on their own,
that we call "symmetry defects" and "reduced QHI", provided the manifolds are
endowed with an additional "non ambiguous structure", a new type of
combinatorial structure that we introduce in this paper. A suitably normalized
version of the symmetry defects applies to compact -manifolds endowed with
-characters, beyond the case of cusped manifolds. Given a
manifold with non empty boundary, we provide a partial "holographic"
description of the non-ambiguous structures in terms of the intrinsic geometric
topology of . Special instances of non ambiguous structures can be
defined by means of taut triangulations, and the symmetry defects have a
particularly nice behaviour on such "taut structures". Natural examples of taut
structures are carried by any mapping torus with punctured fibre of negative
Euler characteristic, or by sutured manifold hierarchies. For a cusped
hyperbolic -manifold which fibres over , we address the question of
determining whether the fibrations over a same fibered face of the Thurston
ball define the same taut structure. We describe a few examples in detail. In
particular, they show that the symmetry defects or the reduced QHI can
distinguish taut structures associated to different fibrations of . To
support the guess that all this is an instance of a general behaviour of state
sum invariants of 3-manifolds based on some theory of 6j-symbols, finally we
describe similar results about reduced Turaev-Viro invariants.Comment: 58 pages, 32 figures; exposition improved, ready for publicatio
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