A surprising fact about the spanning tree entropy for many planar
lattices is that its value is closely related to hyperbolic
geometry. We conjecture sharp upper and lower bounds for the
spanning tree entropy of any planar lattice. These bounds come
from volumes of associated hyperbolic alternating links,
hyperbolic polyhedra and regular ideal bipyramids. We explain the
context and recent progress on our conjecture, which lies at the
intersection of hyperbolic geometry, number theory, probability
and graph theory. This is joint work with Ilya Kofman.
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