Eventually Periodic Resolutions for Groups of Type VFP, Part I

Sean Carroll (Boston, MA)

September 26, 2025 (in-person)

Abstract: We will present a new construction of eventually periodic projective resolutions for modules of type FP-infinity over the integral group rings 𝛤 = G / (c) of type VFP, with G of type FP and c a central element.

Our approach applies more generally to quotients of rings of finite left global dimension by a central regular element. It utilizes a construction of Shamash, combined with the iterated mapping cone technique, to systematically eliminate homology from a complex.

We demonstrate the computational advantage of our method through explicit calculations for several important families of groups:

- amalgamated products of cyclic groups (including SL(2, Z)),

- hyperbolic triangle groups,

- central quotients of Heisenberg groups,

- the mapping class groups of the punctured plane.

The constructed resolutions are of equal or lower rank in each degree compared to known results. In the particular case of SL(2, Z), we produce an eventually 2-periodic resolution of the trivial module, which improves on the 4-periodic resolution constructed earlier by other authors. Our resolutions enable calculations of integral (co)homology to be performed purely algebraically, thereby recovering results originally proved using geometric methods.

Our construction also gives rise to generalized matrix factorizations which, in the commutative local case, recover the original results of Eisenbud without using tools from commutative algebra.

We will discuss further results in this direction in a subsequent talk.