Connes's rigidity conjecture asserted that a countable ICC group with Kazhdan's
property (T) is determined up to isomorphism
by its group von Neumann algebra (Connes 1994,
p. 551). An ICC group is infinite
and has every nonidentity conjugacy class infinite.
Writing 
for the group von Neumann algebra of 
, the conjecture asserted
 |
An AI-generated proof given by OpenAI (2026) disproved the conjecture by constructing finitely generated ICC
groups with Kazhdan's property (T) 
,
,
,
...that are pairwise nonisomorphic and satisfy
 |
Moreover, 
contains a subgroup isomorphic
to 
of index 
, so the groups are mutually
commensurable, meaning that every pair has isomorphic subgroups of finite index.
Popa (2007) proved that the fibers of the group-factor
correspondence are at most countable and later asked
whether the correspondence is finite-to-one on ICC groups
with Kazhdan's property (T) (Popa 2013). The
countably infinite family
gives a negative answer to this finite-to-one question.
-Rigidity." Paris, France, June 2013.