Non-equational stable groups.
Speaker: Rizos Sklinos
Abstract: The notion of equationality has been introduced by Srour and further
developed by Pillay-Srour. It is best understood intuitively as a notion
of Noetherianity on instances of first-order formulas. A first-order
theory is equational when every first-order formula is equivalent to a
boolean combination of equations.
Equationality implies stability and for many years these two notions were
identified, as only an "artificial" example of Hrushovski (a tweaked
pseudo-space) was witnessing otherwise. Recently Sela proved that the
theory of the free group is stable but not equational providing us with
the first natural example of a stable non-equational theory.
We give a transparent proof of the non-equationality of the free group and
we expand the result to the first-order theory of any nontrivial free
product which is not Z_2*Z_2.
In combination with Sela's deep result that the free product of stable
groups is still stable, our result gives an abundance of examples of new
stable non-equational theories.
This is a joint work with Isabel Müller.