$τ$-tilting theory and silting theory of skew group algebra extensions
Let $Λ$ be a finite dimensional algebra with an action by a finite group $G$ and $A:= Λ*G$ the skew group algebra. One of our main results asserts that the canonical restriction-induction adjoint pair of the skew group algebra extension $Λ\subset A$ induces a poset isomorphism between the poset of $G$-stable support $τ$-tilting modules over $Λ$ and that of $(\!\!\!\mod G)$-stable support $τ$-tilting modules over $A$. We also establish a similar poset isomorphism of posets of appropriate classes of silting complexes over $Λ$ and $A$. These two results generalize and unify preceding results by Huang-Zhang, Breaz-Marcus-Modoi and the second and the third authors. Moreover, we give a practical condition under which $τ$-tilting finiteness and silting discreteness of $Λ$ are inherited to those of $A$. As applications we study $τ$-tilting theory and silting theory of the (generalized) preprojective algebras and the folded mesh algebras. Among other things, we determine the posets of support $τ$-tilting modules and of silting complexes over preprojective algebra $Π(\Bbb{L}_{n})$ of type $\Bbb{L}_{n}$.