We study possible values of the global dimension of endomorphism algebras of 2-term silting complexes. We show that for any algebra A whose global dimension gl.dim A ≤ 2 and any 2-term silting complex P in the bounded derived category D^b(A) of A, the global dimension of End_{D^b(A)}(P) is at most 7. We also show that for each n > 2, there is an algebra A with gl.dim A = n such that D^b(A) admits a 2-term silting complex P with gl.dim End_{D^b(A)}(P) infinite.

We introduce and study mutation of torsion pairs, as a generalization of mutation of cluster tilting objects, rigid objects and maximal rigid objects. It is proved that any mutation of a torsion pair is again a torsion pair. A geometric realization of mutation of torsion pairs in the cluster category of type A_n or A_\infty is given via rotation of Ptolemy diagrams.

In this paper, we give a complete classification of cotorsion pairs in a cluster category C of type A^\infty_\infty via certain configurations of arcs, called τ-compact Ptolemy diagrams, in an infinite strip with marked points. As applications, we classify t-structures and functorially finite rigid subcategories in C, respectively. We also deduce Liu-Paquette's classification of cluster tilting subcategories of C and Ng's classification of torsion pairs in the cluster category of type A^\infty_\infty.

We give a finite presentation for the braid twist group of a decorated surface. If the decorated surface arises from a triangulated marked surface without punctures, we obtain a finite presentation for the spherical twist group of the associated 3-Calabi–Yau triangulated category. The motivation/application is that the result will be used to show that the (principal component of) space of stability conditions on the 3-Calabi–Yau category is simply connected in the sequel [King and Qiu, Invent. Math., to appear].

We study the Ginzburg dg algebra Γ_T associated with the quiver with potential arising from a triangulation T of a decorated marked surface S_△⁠, in the sense of [22]. We show that there is a canonical way to identify all finite-dimensional derived categories D_{fd}(Γ_T)⁠, denoted by D_{fd}(S_△)⁠. As an application, we show that the spherical twist group ST(S_△) associated with D_{fd}(S_△) acts faithfully on its space of stability conditions.