On τ-tilting finite simply connected algebras

We show that τ -tilting finite simply connected algebras are representation-finite. Then, some related algebras are considered, including iterated tilted algebras, tubular algebras and so on. We also prove that the τ -tilting finiteness of non-sincere algebras can be reduced to that of sincere algebras. This motivates us to give a complete list of τ -tilting finite sincere simply connected algebras.