$\tau$-tilting finiteness of biserial algebras

In this paper we treat the $\tau$-tilting finiteness of biserial (respectively special biserial) algebras over algebraically closed (respectively arbitrary) fields. Inside these families, to compare the notions of representation-finiteness and $\tau$-tilting finiteness, we reduce the problem to the $\tau$-tilting finiteness of minimal representation-infinite (special) biserial algebras. Building upon the classification of minimal representation-infinite algebras, we fully determine which minimal representation-infinite (special) biserial algebras are $\tau$-tilting finite and which ones are not. To do so, we use the brick-$\tau$-rigid correspondence of Demonet, Iyama and Jasso, and the classification of minimal representation-infinite special biserial algebras due to Ringel. Furthermore, we introduce the notion of minimal $\tau$-tilting infinite algebras, analogous to the notion of minimal representation infinite algebras, and prove that a minimal representation-infinite (special) biserial algebra is minimal $\tau$-tilting infinite if and only if it is a gentle algebra. As a consequence, we conclude that a gentle algebra is $\tau$-tilting infinite if and only if it is representation infinite. We also show that for every minimal representation-infinite (special) biserial algebra, the notions of tilting finiteness and $\tau$-tilting finiteness are equivalent. This implies that a mild (special) biserial algebra is tilting finite if and only if it is brick finite.