Geodesic
An estimate from above of spectral radii of random walks on surface groups
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part II, Tome 240 (1997), pp. 154-165

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Using Gabber's Lemma, we get new estimates of the spectral radius of the simple random walk on the fundamental group of the orientable closed surface of genus $g$, $g\ge2$. In order to get better numerical estimates we base our method on Cannon's classification of the group elements by their cone types. The method may as well be applied to many other groups and graphs with finite numbers of cone types. 
@article{ZNSL_1997_240_a11,
     author = {T. V. Nagnibeda},
     title = {An estimate from above of spectral radii of random walks on surface groups},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {154--165},
     publisher = {mathdoc},
     volume = {240},
     year = {1997},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1997_240_a11/}
}
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T. V. Nagnibeda. An estimate from above of spectral radii of random walks on surface groups. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part II, Tome 240 (1997), pp. 154-165. http://geodesic.mathdoc.fr/item/ZNSL_1997_240_a11/