We show that for three dimensional gravity with higher genus boundary
conditions, if the theory possesses a sufficiently light scalar, there is a
second order phase transition where the scalar field condenses. This three
dimensional version of the holographic superconducting phase transition occurs
even though the pure gravity solutions are locally AdS3. This is in addition
to the first order Hawking-Page-like phase transitions between different
locally AdS3 handlebodies. This implies that the R\'enyi entropies of
holographic CFTs will undergo phase transitions as the R\'enyi parameter is
varied, as long as the theory possesses a scalar operator which is lighter than
a certain critical dimension. We show that this critical dimension has an
elegant mathematical interpretation as the Hausdorff dimension of the limit set
of a quotient group of AdS3, and use this to compute it, analytically near
the boundary of moduli space and numerically in the interior of moduli space.
We compare this to a CFT computation generalizing recent work of Belin, Keller
and Zadeh, bounding the critical dimension using higher genus conformal blocks,
and find a surprisingly good match
