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European Mathematical Society Publishing House
2024-03-28 17:27:49
5
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=GGD&vol=1&iss=2&update_since=2024-03-28
Groups, Geometry, and Dynamics
Groups Geom. Dyn.
GGD
1661-7207
1661-7215
Group theory and generalizations
10.4171/GGD
http://www.ems-ph.org/doi/10.4171/GGD
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
1
2007
2
Denominator bounds in Thompson-like groups and flows
Danny
Calegari
California Institute of Technology, PASADENA, UNITED STATES
Thompson's group, rotation number, rationality
Let T denote Thompson's group of piecewise 2-adic linear homeomorphisms of the circle. Ghys and Sergiescu showed that the rotation number of every element of T is rational, but their proof is very indirect. We give here a short, direct proof using train tracks, which generalizes to elements of PL+(S1) with rational break points and derivatives which are powers of some fixed integer, and also to certain flows on surfaces which we call Thompson-like. We also obtain an explicit upper bound on the smallest period of a fixed point in terms of data which can be read off from the combinatorics of the homeomorphism.
Dynamical systems and ergodic theory
General
101
109
10.4171/GGD/6
http://www.ems-ph.org/doi/10.4171/GGD/6
Représentation par des transvections des groupes d'Artin–Tits
Eddy
Godelle
Université de Caen, CAEN CEDEX, FRANCE
Representation of braid groups, automorphisms of free groups, episturmian morphisms
In a recent article, C. Kassel and C. Reutenauer studied the connection between the 4 strand braid group and Sturmian morphisms in word combinatorics. The aim of the current work is to extend this approach into a general connection between braid groups (of any index) and episturmian morphisms, a natural generalization of sturmian morphisms. Our key tool consists in associating with every graph a certain finite family of automorphisms of a free group. In the case of a complete graph, we recover some well-known family of episturmian morphisms. Now, considering the path of length n, we deduce a seemingly new representation of the braid group Bn +1 in Aut(Fn). By considering some other graphs, we similarly obtain representations of various Artin–Tits groups, in particular some affine braid groups. Our representation is faithful for B3 and B4; for other cases, the question of faithfulness remains open.
Group theory and generalizations
Computer science
General
111
133
10.4171/GGD/7
http://www.ems-ph.org/doi/10.4171/GGD/7
Deformation spaces of trees
Vincent
Guirardel
Université de Rennes 1, RENNES CEDEX, FRANCE
Gilbert
Levitt
Université de Caen Basse-Normandie, CAEN CEDEX, FRANCE
Deformation space, actions on trees, contractible
Let G be a finitely generated group. Two simplicial G-trees are said to be in the same deformation space if they have the same elliptic subgroups (if H fixes a point in one tree, it also does in the other). Examples include Culler–Vogtmann's outer space and spaces of JSJ decompositions. We discuss what features are common to trees in a given deformation space, how to pass from one tree to all other trees in its deformation space, and the topology of deformation spaces. In particular, we prove that all deformation spaces are contractible complexes.
Group theory and generalizations
General
135
181
10.4171/GGD/8
http://www.ems-ph.org/doi/10.4171/GGD/8
Connectedness at infinity of systolic complexes and groups
Damian
Osajda
Uniwersytet Wrocławski, WROCŁAW, POLAND
Simplicial non-positive curvature, topology at infinity
By studying connectedness at infinity of systolic groups we distinguish them from some other classes of groups, in particular from the fundamental groups of manifolds covered by Euclidean space of dimension at least three. We also study semistability at infinity for some systolic groups.
Group theory and generalizations
Manifolds and cell complexes
General
183
203
10.4171/GGD/9
http://www.ems-ph.org/doi/10.4171/GGD/9
A short note on Dynkin groups and convergence groups
Asli
Yaman
, BELLATERRA, SPAIN
Dynkin groups, convergence groups
We prove that the family of discrete non-elementary Dynkin groups coincides with the family of non-elementary convergence groups.
Group theory and generalizations
General
205
208
10.4171/GGD/10
http://www.ems-ph.org/doi/10.4171/GGD/10