1,898 research outputs found
The normal closure of big Dehn twists, and plate spinning with rotating families
We study the normal closure of a big power of one or several Dehn twists in a
Mapping Class Group. We prove that it has a presentation whose relators
consists only of commutators between twists of disjoint support, thus answering
a question of Ivanov. Our method is to use the theory of projection complexes
of Bestvina Bromberg and Fujiwara, together with the theory of rotating
families, simultaneously on several spaces.Comment: 32 page
Combination of convergence groups
We state and prove a combination theorem for relatively hyperbolic groups
seen as geometrically finite convergence groups. For that, we explain how to
contruct a boundary for a group that is an acylindrical amalgamation of
relatively hyperbolic groups over a fully quasi-convex subgroup. We apply our
result to Sela's theory on limit groups and prove their relative hyperbolicity.
We also get a proof of the Howson property for limit groups.Comment: Published in Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol7/paper27.abs.htm
Detecting free splittings in relatively hyperbolic groups
We describe an algorithm which determines whether or not a group which is
hyperbolic relative to abelian groups admits a nontrivial splitting over a
finite group.Comment: 15 pages. Version 2 is 17 pages, edited in light of referee's
comments. To appear in Transactions of the AM
Deciding Isomorphy using Dehn fillings, the splitting case
We solve Dehn's isomorphism problem for virtually torsion-free relatively
hyperbolic groups with nilpotent parabolic subgroups.
We do so by reducing the isomorphism problem to three algorithmic problems in
the parabolic subgroups, namely the isomorphism problem, separation of torsion
(in their outer automorphism groups) by congruences, and the mixed Whitehead
problem, an automorphism group orbit problem. The first step of the reduction
is to compute canonical JSJ decompositions. Dehn fillings and the given
solutions of the algorithmic problems in the parabolic groups are then used to
decide if the graphs of groups have isomorphic vertex groups and, if so,
whether a global isomorphism can be assembled.
For the class of finitely generated nilpotent groups, we give solutions to
these algorithmic problems by using the arithmetic nature of these groups and
of their automorphism groups.Comment: 76 pages. This version incorporates referee comments and corrections.
The main changes to the previous version are a better treatment of the
algorithmic recognition and presentation of virtually cyclic subgroups and a
new proof of a rigidity criterion obtained by passing to a torsion-free
finite index subgroup. The previous proof relied on an incorrect result. To
appear in Inventiones Mathematica
Copies of a one-ended group in a Mapping Class Group
We establish that, given a compact orientable surface, and a
finitely presented one-ended group, the set of copies of in the mapping
class group consisting of only pseudo-anosov elements
except identity, is finite up to conjugacy. This relies on a result of Bowditch
on the same problem for images of surfaces groups. He asked us whether we could
reduce the case of one-ended groups to his result ; this is a positive answer.
Our work involves analogues of Rips and Sela canonical cylinders in curve
complexes, and the argument of Delzant to bound the number of images of a
one-ended group in a hyperbolic group.Comment: 16 pages, 3 figures, revise
Spectral theorems for random walks on mapping class groups and
We establish spectral theorems for random walks on mapping class groups of
connected, closed, oriented, hyperbolic surfaces, and on . In
both cases, we relate the asymptotics of the stretching factor of the
diffeomorphism/automorphism obtained at time of the random walk to the
Lyapunov exponent of the walk, which gives the typical growth rate of the
length of a curve -- or of a conjugacy class in -- under a random product
of diffeomorphisms/automorphisms.
In the mapping class group case, we first observe that the drift of the
random walk in the curve complex is also equal to the linear growth rate of the
translation lengths in this complex. By using a contraction property of typical
Teichm\"uller geodesics, we then lift the above fact to the realization of the
random walk on the Teichm\"uller space. For the case of , we
follow the same procedure with the free factor complex in place of the curve
complex, and the outer space in place of the Teichm\"uller space. A general
criterion is given for making the lifting argument possible.Comment: 45 pages, 3 figures. arXiv admin note: text overlap with
arXiv:1506.0724
Symbolic dynamics and relatively hyperbolic groups
We study the action of a relatively hyperbolic group on its boundary, by
methods of symbolic dynamics. Under a condition on the parabolic subgroups, we
show that this dynamical system is finitely presented. We give examples where
this condition is satisfied, including geometrically finite kleinian groups.Comment: Revision, 16 pages, 1 figur
Existential questions in (relatively) hyperbolic groups {\it and} Finding relative hyperbolic structures
This arXived paper has two independant parts, that are improved and corrected
versions of different parts of a single paper once named "On equations in
relatively hyperbolic groups".
The first part is entitled "Existential questions in (relatively) hyperbolic
groups". We study there the existential theory of torsion free hyperbolic and
relatively hyperbolic groups, in particular those with virtually abelian
parabolic subgroups. We show that the satisfiability of systems of equations
and inequations is decidable in these groups.
In the second part, called "Finding relative hyperbolic structures", we
provide a general algorithm that recognizes the class of groups that are
hyperbolic relative to abelian subgroups.Comment: Two independant parts 23p + 9p, revised. To appear separately in
Israel J. Math, and Bull. London Math. Soc. respectivel
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