23 research outputs found
Hyperbolic towers and independent generic sets in the theory of free groups
We use hyperbolic towers to answer some model theoretic questions around the
generic type in the theory of free groups. We show that all the finitely
generated models of this theory realize the generic type , but that there
is a finitely generated model which omits . We exhibit a finitely
generated model in which there are two maximal independent sets of realizations
of the generic type which have different cardinalities. We also show that a
free product of homogeneous groups is not necessarily homogeneous.Comment: to appear in Proceedings of the conference "Recent developments in
Model Theory", Notre Dame Journal of Formal Logi
Zariski Closures and Subgroup Separability
The main result of this article is a refinement of the well-known subgroup
separability results of Hall and Scott for free and surface groups. We show
that for any finitely generated subgroup, there is a finite dimensional
representation of the free or surface group that separates the subgroup in the
induced Zariski topology. As a corollary, we establish a polynomial upper bound
on the size of the quotients used to separate a finitely generated subgroup in
a free or surface group.Comment: Final version. To appear in Selecta Mat
Krull dimension for limit groups II: aligning JSJ decompositions
This is the second paper in a sequence on Krull dimension for limit groups,
answering a question of Z. Sela. In it we develop the notion of a resolution of
a sequence of limit groups and show how to derive resolutions of low complexity
from resolutions of high complexity.Comment: Consistent with other papers in the series, I hope. Not a major
rewrite. The statement of Theorem 1.4 is consistent with paper 4 (the new
one) now. (And they have the same numbering, coincidentally.