The singularity and cosingularity categories of C ∗ B G for groups with cyclic Sylow p-subgroups

We construct a differential graded algebra (DGA) modelling certain A∞ algebras associated with a finite group G with cyclic Sylow subgroups, namely H∗BG and H∗ΩBG∧p. We use our construction to investigate the singularity and cosingularity categories of these algebras. We give a complete classification of the indecomposables in these categories, and describe the Auslander--Reiten quiver. The theory applies to Brauer tree algebras in arbitrary characteristic, and we end with an example in characteristic zero coming from the Hecke algebras of symmetric groups