Geodesic
Bounded discrete holomorphic functions on the hyperbolic plane
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Topology and physics, Tome 302 (2018), pp. 202-213

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It is shown that, for the discretization of complex analysis introduced earlier by S. P. Novikov and the present author, there exists a rich family of bounded discrete holomorphic functions on the hyperbolic (Lobachevsky) plane endowed with a triangulation by regular triangles whose vertices have even valence. Namely, it is shown that every discrete holomorphic function defined in a bounded convex domain can be extended to a bounded discrete holomorphic function on the whole hyperbolic plane so that the Dirichlet energy be finite. 
@article{TM_2018_302_a8,
     author = {I. A. Dynnikov},
     title = {Bounded discrete holomorphic functions on the hyperbolic plane},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {202--213},
     publisher = {mathdoc},
     volume = {302},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2018_302_a8/}
}
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I. A. Dynnikov. Bounded discrete holomorphic functions on the hyperbolic plane. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Topology and physics, Tome 302 (2018), pp. 202-213. http://geodesic.mathdoc.fr/item/TM_2018_302_a8/