On the Picard group of the stable module category for infinite groups

We introduce the stable module $\mathrm{\infty }$-category for groups of type $\mathrm{\Phi }$ as an enhancement of the stable category defined by N. Mazza and P. Symonds. For groups of type $\mathrm{\Phi }$ which act on a tree, we show that the stable module $\mathrm{\infty }$-category decomposes in terms of the associated graph of groups. For groups which admit a finite-dimensional cocompact model for the classifying space for proper actions, we exhibit a decomposition in terms of the stable module $\mathrm{\infty }$-categories of their finite subgroups. We use these decompositions to provide methods to compute the Picard group of the stable module category. In particular, we provide a description of the Picard group for countable locally finite $p$-groups.