Determinants of g-tameness for group algebras and their blocks

Determine what structural criteria or invariants decide g-tameness for group algebras of finite groups and for their blocks; that is, characterize when the union of g-cones of two-term silting complexes is dense in the Grothendieck group vector space for such algebras.

Background

The paper reviews relationships among representation type, τ-tilting finiteness, and g-tameness, noting that tame algebras are g-tame and tame blocks are τ-tilting finite. However, unlike representation type (classified via defect groups), there is no known structural criterion for g-tameness of group algebras or their blocks.

The authors explicitly point out that nothing is currently known about which features of a finite group or its subgroups govern g-tameness, motivating a search for analogues of defect groups or p-hyperfocal subgroups in this context.