86,134 research outputs found
(Non)Commutative Hopf algebras of trees and (quasi)symmetric functions
The Connes-Kreimer Hopf algebra of rooted trees, its dual, and the Foissy
Hopf algebra of of planar rooted trees are related to each other and to the
well-known Hopf algebras of symmetric and quasi-symmetric functions via a pair
of commutative diagrams. We show how this point of view can simplify
computations in the Connes-Kreimer Hopf algebra and its dual, particularly for
combinatorial Dyson-Schwinger equations.Comment: For March 2006 CIRM conference "Renormalization and Galois theories
Limit groups and groups acting freely on R^n-trees
We give a simple proof of the finite presentation of Sela's limit groups by
using free actions on R^n-trees. We first prove that Sela's limit groups do
have a free action on an R^n-tree. We then prove that a finitely generated
group having a free action on an R^n-tree can be obtained from free abelian
groups and surface groups by a finite sequence of free products and
amalgamations over cyclic groups. As a corollary, such a group is finitely
presented, has a finite classifying space, its abelian subgroups are finitely
generated and contains only finitely many conjugacy classes of non-cyclic
maximal abelian subgroups.Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol8/paper39.abs.htm
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