Geodesic
Actions of Finitely Generated Groups and Semigroups on a Plane by Means of Isometries
Matematičeskie zametki, Tome 75 (2004) no. 4, pp. 523-548.

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The problem of uniform distribution of orbits on a plane for actions of finitely generated groups of isometries is studied. A criterion for every orbit to be dense is presented. The asymptotics for the frequency with which an orbit hits a compact set is obtained. 
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Ya. B. Vorobets. Actions of Finitely Generated Groups and Semigroups on a Plane by Means of Isometries. Matematičeskie zametki, Tome 75 (2004) no. 4, pp. 523-548. http://geodesic.mathdoc.fr/item/MZM_2004_75_4_a3/

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