106,885 research outputs found
Rips Induction: Index of the dual lamination of an -tree
Let be a -tree in the boundary of the Outer Space CV, with dense
orbits. The -index of is defined by means of the dual lamination of .
It is a generalisation of the Euler-Poincar\'e index of a foliation on a
surface. We prove that the -index of is bounded above by , and we
study the case of equality. The main tool is to develop the Rips Machine in
order to deal with systems of isometries on compact -trees. Combining our
results on the \CQ-index with results on the classical geometric index of a
tree, we obtain a beginning of classification of trees. As a consequence, we
give a classification of iwip outer automorphisms of the free group, by
discussing the properties of their attracting and repelling trees.Comment: 33 pages. The previous version has been splitted in two disjoint
papers. See also Botanic of irreducible automorphisms of free group
A Combinatorial Analog of a Theorem of F.J.Dyson
Tucker's Lemma is a combinatorial analog of the Borsuk-Ulam theorem and the
case n=2 was proposed by Tucker in 1945. Numerous generalizations and
applications of the Lemma have appeared since then. In 2006 Meunier proved the
Lemma in its full generality in his Ph.D. thesis. There are generalizations and
extensions of the Borsuk-Ulam theorem that do not yet have combinatorial
analogs. In this note, we give a combinatorial analog of a result of Freeman J.
Dyson and show that our result is equivalent to Dyson's theorem. As with
Tucker's Lemma, we hope that this will lead to generalizations and applications
and ultimately a combinatorial analog of Yang's theorem of which both
Borsuk-Ulam and Dyson are special cases.Comment: Original version: 7 pages, 2 figures. Revised version: 12 pages, 4
figures, revised proofs. Final revised version: 9 pages, 2 figures, revised
proof
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