  We consider pure three-dimensional quantum gravity with a negative
cosmological constant. The sum of known contributions to the partition function
from classical geometries can be computed exactly, including quantum
corrections. However, the result is not physically sensible, and if the model
does exist, there are some additional contributions. One possibility is that
the theory may have long strings and a continuous spectrum. Another possibility
is that complex geometries need to be included, possibly leading to a
holomorphically factorized partition function. We analyze the subleading
corrections to the Bekenstein-Hawking entropy and show that these can be
correctly reproduced in such a holomorphically factorized theory. We also
consider the Hawking-Page phase transition between a thermal gas and a black
hole and show that it is a phase transition of Lee-Yang type, associated with a
condensation of zeros in the complex temperature plane. Finally, we analyze
pure three-dimensional supergravity, with similar results.
