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On the isomorphism of Richard Thompson’s group with the Ptolemy group. (Sur l’isomorphisme du groupe de Richard Thompson avec le groupe de Ptolémée.) (French) Zbl 0911.20031
Schneps, Leila (ed.) et al., Geometric Galois actions. 2. The inverse Galois problem, moduli spaces and mapping class groups. Proceedings of the conference on geometry and arithmetic of moduli spaces, Luminy, France, August 1995. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 243, 313-324 (1997).
The author gives a demonstration of the following observation of M. Kontsevich: The universal Ptolemy group $$G$$ defined by R. Penner is isomorphic with Richard Thompson’s homeomorphism group $$\mathbf{PL}_2(S^1)$$. That there is a connection between the two groups follows from the following observations: Penner considers an action on tessellations of the hyperbolic plane by ideal triangles which replaces a two-triangle quadrilateral by switching the dividing diagonal; the Ptolemy group consists of equivalence classes of finite compositions of such motions. Thompson’s group elements are represented by pairs of finite binary trees; each such tree is, via duality, a partial dual to an ideal triangular tesselation; and the group element can be considered as transition from the first tree (tesselation) to the second.
For the entire collection see [Zbl 0868.00040].
Reviewer: J.W.Cannon (Provo)

##### MSC:
 20F65 Geometric group theory 30F60 Teichmüller theory for Riemann surfaces 20D08 Simple groups: sporadic groups 57S20 Noncompact Lie groups of transformations 57S10 Compact groups of homeomorphisms 57S05 Topological properties of groups of homeomorphisms or diffeomorphisms