The Balmer spectrum and telescope conjecture for infinite groups

We determine the Balmer spectrum of dualisable objects in the stable module category for ${\mathrm{H}}_1\mathfrak{F}$ groups of type ${\mathrm{F}\mathrm{P}}_{\mathrm{\infty }}$ and show that the telescope conjecture holds for these categories. We also determine the spectrum of dualisable objects for certain infinite free products of finite groups. Using this, we give examples where the stable category is not stratified by the spectrum of dualisable objects and where the telescope conjecture does not hold.