Superrosy dependent groups having finitely satisfiable generics

We study a model theoretic context (finite thorn rank, NIP, with finitely satisfiable generics) which is a common generalization of groups of finite Morley rank and definably compact groups in o-minimal structures. We show that assuming thorn rank 1, the group is abelian-by-finite, and assuming thorn rank 2 the group is solvable by finite. Also a field is algebraically closed

On \omega-categorical, generically stable groups

We prove that each \omega-categorical, generically stable group is solvable-by-finite.Comment: 11 page

Dimensional groups and fields

We shall define a general notion of dimension, and study groups and rings whose interpretable sets carry such a dimensio. In particular, we deduce chain conditions for groups, definability results for fields and domains, and show that pseudofinite groups contain big finite-by-abelian subgroups, and pseudofinite groups of dimension 2 contain big soluble subgroups

Stable embeddedness and NIP

We give sufficient conditions for a predicate P in a complete theory T to be stably embedded: P with its induced 0-definable structure has "finite rank", P has NIP in T and P is 1-stably embedded. This generalizes recent work by Hasson and Onshuus in the case where P is o-minimal in T.Comment: 10 page

Definable Envelopes of Nilpotent Subgroups of Groups with Chain Conditions on Centralizers

An $\mathfrak{M}_C$ group is a group in which all chains of centralizers have finite length. In this article, we show that every nilpotent subgroup of an $\mathfrak{M}_C$ group is contained in a definable subgroup which is nilpotent of the same nilpotence class. Definitions are uniform when the lengths of chains are bounded

Sets, groups, and fields definable in vector spaces with a bilinear form

We study definable sets, groups, and fields in the theory $T_\infty$ of infinite-dimensional vector spaces over an algebraically closed field equipped with a nondegenerate symmetric (or alternating) bilinear form. First, we define an ($\mathbb{N}\times \mathbb{Z},\leq_{lex}$)-valued dimension on definable sets in $T_\infty$ enjoying many properties of Morley rank in strongly minimal theories. Then, using this dimension notion as the main tool, we prove that all groups definable in $T_\infty$ are (algebraic-by-abelian)-by-algebraic, which, in particular, answers a question of Granger. We conclude that every infinite field definable in $T_\infty$ is definably isomorphic to the field of scalars of the vector space. We derive some other consequences of good behaviour of the dimension in $T_\infty$, e.g. every generic type in any definable set is a definable type; every set is an extension base; every definable group has a definable connected component. We also consider the theory $T^{RCF}_\infty$ of vector spaces over a real closed field equipped with a nondegenerate alternating bilinear form or a nondegenerate symmetric positive-definite bilinear form. Using the same construction as in the case of $T_\infty$, we define a dimension on sets definable in $T^{RCF}_\infty$, and using it we prove analogous results about definable groups and fields: every group definable in $T^{RCF}_{\infty}$ is (semialgebraic-by-abelian)-by-semialgebraic (in particular, it is (Lie-by-abelian)-by-Lie), and every field definable in $T^{RCF}_{\infty}$ is definable in the field of scalars, hence it is either real closed or algebraically closed.Comment: v2: The particular bounds on dimension obtained in Section 3 were corrected, and a number of minor corrections has been made throughout the pape