Geodesic
On the uniform distribution of orbits of finitely generated groups and semigroups of plane isometries
Sbornik. Mathematics, Tome 195 (2004) no. 2, pp. 163-186
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Actions by isometries on a Euclidean plane of free groups and free semigroups with arbitrarily many generators and of free products of groups of order 2 are considered. It is shown that, in the typical case, all orbits of the action are uniformly distributed in the plane. The actions for which the distribution of orbits fails to be uniform are explicitly described. 
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Ya. B. Vorobets. On the uniform distribution of orbits of finitely generated groups and semigroups of plane isometries. Sbornik. Mathematics, Tome 195 (2004) no. 2, pp. 163-186. http://geodesic.mathdoc.fr/item/SM_2004_195_2_a1/

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