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Add to each level of binary tree edges to make the induced graph on the level a uniform expander.
V. A. Kaimanovich and A. M. Vershik, Random walks on discrete groups: boundary and entropy. Ann. Probab
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I. Benjamini and J. Cao, Examples of simply-connected Liouville manifolds with positive spectrum. Jour. Diff. Geom. and Appl
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I. Benjamini and O. Schramm, Harmonic functions on planar graphs and almost planar manifolds via circle packings. Inventiones Math
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S. Hoory, N. Linial and A. Wigderson, Exapnader graphs and their applications. Bull. A.M.S
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I. Benjamini and D. Revelle, Instability of Set Recurrence and Green’s Function on Groups with the Liouville Property. Potential analysis
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B. Kleiner, A new proof of Gromov’s theorem on groups of polynomial growth. J. Amer. Math. Soc
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