In this weeks Theory Seminar, Emmanuel, from the Université de Cétreil, will present his work on ‘Residual properties of groups in the continuous Weihrauch lattice.

For every class C of groups, one can define a sort of projection onto C: indeed, every group has a greatest residually C quotient, where a group is residually C if every non-identity element in this group has a non-identity image via a morphism to a group in C.

I study how discontinuous this projection map is, in terms of continuous Weihrauch reducibility over the topology of the space of marked groups. This provides a useful complement to the Borel Classification of C: indeed, the complexity of studying homomorphisms towards groups of C is much better captured by this classification than by the usual Borel classification of C.

I will introduce the main notions that are relevant in this context: quasi-varieties, equational noetherianity, INIP groups.