Maximal Finite-Set Stabilizers in Thompson's Group T

Let \(\D=\Z[1/2]/\Z\) be the dyadic circle and let \(T\) be Thompson's orientation-preserving circle group. We prove that, for every nonempty finite set \(A\subset\D\) , its setwise stabilizer is a maximal proper subgroup of countably infinite index in \(T\) . If \(|A|=k\) , then \[ \Stab_T(A)\cong F^k\rtimes C_k=F\wr C_k, \] where the cyclic group permutes the factors. Stabilizers with the same cardinality are conjugate, whereas those with distinct cardinalities are pairwise nonisomorphic. The proof establishes a general circular-order criterion: a group with the finite circular extension property acts primitively on the \(k\) -element subsets for every \(k\) . This yields a countably infinite family of maximal infinite-index subgroups and supplies examples requested in the 2024 AIM problem list on groups of dynamical origin. The theorem is apparently unrecorded in the literature checked; no absolute priority claim is made.