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.