20,153 research outputs found
A geometric model of tube categories
We give a geometric model for a tube category in terms of homotopy classes of
oriented arcs in an annulus with marked points on its boundary. In particular,
we interpret the dimensions of extension groups of degree 1 between
indecomposable objects in terms of negative geometric intersection numbers
between corresponding arcs, giving a geometric interpretation of the
description of an extension group in the cluster category of a tube as a
symmetrized version of the extension group in the tube. We show that a similar
result holds for finite dimensional representations of the linearly oriented
quiver of type A-double-infinity.Comment: 15 pages, 7 figures. Discussion of maximal rigid objects and
triangulations at end of Section 3. Minor correction
Recovering the topology of surfaces from cluster algebras
We present an effective method for recovering the topology of a bordered
oriented surface with marked points from its cluster algebra. The information
is extracted from the maximal triangulations of the surface, those that have
exchange quivers with maximal number of arrows in the mutation class. The
method gives new proofs of the automorphism and isomorphism problems for the
surface cluster algebras, as well as the uniqueness of the
Fomin-Shapiro-Thurston block decompositions of the exchange quivers of the
surface cluster algebras. The previous proofs of these results followed a
different approach based on Gu's direct proof of the last result. The method
also explains the exceptions to these results due to pathological problems with
the maximal triangulations of several surfaces.Comment: 29 pages, AMS Late
Crossings and nestings in set partitions of classical types
In this article, we investigate bijections on various classes of set
partitions of classical types that preserve openers and closers. On the one
hand we present bijections that interchange crossings and nestings. For types B
and C, they generalize a construction by Kasraoui and Zeng for type A, whereas
for type D, we were only able to construct a bijection between non-crossing and
non-nesting set partitions. On the other hand we generalize a bijection to type
B and C that interchanges the cardinality of the maximal crossing with the
cardinality of the maximal nesting, as given by Chen, Deng, Du, Stanley and Yan
for type A. Using a variant of this bijection, we also settle a conjecture by
Soll and Welker concerning generalized type B triangulations and symmetric fans
of Dyck paths.Comment: 22 pages, 7 Figures, removed erroneous commen
Topological invariants for semigroups of holomorphic self-maps of the unit disc
Let , be two one-parameter semigroups of holomorphic
self-maps of the unit disc . Let be a homeomorphism. We prove that, if for all , then extends to a homeomorphism of
outside exceptional maximal contact arcs (in particular, for
elliptic semigroups, extends to a homeomorphism of ).
Using this result, we study topological invariants for one-parameter semigroups
of holomorphic self-maps of the unit disc.Comment: 28 pages, final version, to appear in J. Math. Pures App
Solutions of Word Equations over Partially Commutative Structures
We give NSPACE(n log n) algorithms solving the following decision problems.
Satisfiability: Is the given equation over a free partially commutative monoid
with involution (resp. a free partially commutative group) solvable?
Finiteness: Are there only finitely many solutions of such an equation? PSPACE
algorithms with worse complexities for the first problem are known, but so far,
a PSPACE algorithm for the second problem was out of reach. Our results are
much stronger: Given such an equation, its solutions form an EDT0L language
effectively representable in NSPACE(n log n). In particular, we give an
effective description of the set of all solutions for equations with
constraints in free partially commutative monoids and groups
Commensurations of the Johnson kernel
Let K be the subgroup of the extended mapping class group, Mod(S), generated
by Dehn twists about separating curves. Assuming that S is a closed, orientable
surface of genus at least 4, we confirm a conjecture of Farb that Comm(K),
Aut(K) and Mod(S) are all isomorphic. More generally, we show that any
injection of a finite index subgroup of K into the Torelli group I of S is
induced by a homeomorphism. In particular, this proves that K is co-Hopfian and
is characteristic in I. Further, we recover the result of Farb and Ivanov that
any injection of a finite index subgroup of I into I is induced by a
homeomorphism. Our method is to reformulate these group theoretic statements in
terms of maps of curve complexes.Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol8/paper37.abs.htm
Nielsen equivalence in a class of random groups
We show that for every there exists a torsion-free one-ended
word-hyperbolic group of rank admitting generating -tuples
and such that the -tuples
are not
Nielsen-equivalent in . The group is produced via a probabilistic
construction.Comment: 34 pages, 2 figures; a revised final version, to appear in the
Journal of Topolog
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