3,678 research outputs found
Lone Axes in Outer Space
Handel and Mosher define the axis bundle for a fully irreducible outer
automorphism in "Axes in Outer Space." In this paper we give a necessary and
sufficient condition for the axis bundle to consist of a unique periodic fold
line. As a consequence, we give a setting, and means for identifying in this
setting, when two elements of an outer automorphism group have
conjugate powers.Comment: Significant revisions for the sake of readability, several arguments
filled i
On Groupoids and Hypergraphs
We present a novel construction of finite groupoids whose Cayley graphs have
large girth even w.r.t. a discounted distance measure that contracts
arbitrarily long sequences of edges from the same colour class (sub-groupoid),
and only counts transitions between colour classes (cosets). These groupoids
are employed towards a generic construction method for finite hypergraphs that
realise specified overlap patterns and avoid small cyclic configurations. The
constructions are based on reduced products with groupoids generated by the
elementary local extension steps, and can be made to preserve the symmetries of
the given overlap pattern. In particular, we obtain highly symmetric, finite
hypergraph coverings without short cycles. The groupoids and their application
in reduced products are sufficiently generic to be applicable to other
constructions that are specified in terms of local glueing operations and
require global finite closure.Comment: Explicit completion of H in HxI (Section 2) is unstable (incompatible
with restrictions), hence does not support inductive construction towards
Prop. 2.17 based on Lem 2.16 as claimed. For corresponding technical result,
now see arxiv:1806.08664; for discussion of main applications first announced
here, now see arxiv:1709.0003
Mapping tori of free group automorphisms are coherent
The mapping torus of an endomorphism \Phi of a group G is the HNN-extension
G*_G with bonding maps the identity and \Phi. We show that a mapping torus of
an injective free group endomorphism has the property that its finitely
generated subgroups are finitely presented and, moreover, these subgroups are
of finite type.Comment: 17 pages, published versio
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