Model-Theoretic Definable Binding Groups

• Model-Theoretic Definable Binding Groups are invariants in definable groups that capture the failure of definable connectedness and the complexity of group orbits using canonical type-definable subgroups.
• They provide a model-theoretic analogue to classical topological dynamics by realizing universal definable compactifications and ambits through logical structures like G*/(G*)00 and S_G(M).
• The framework bridges definable dynamics with topological group theory, linking amenability and extreme amenability to properties of minimal subflows and semigroup operations in type spaces.

A model-theoretic definable binding group is a group-theoretic invariant arising in the context of a structure $M$ and a definable group $G \subseteq M^n$, used to analyze compactifications, flows, and topological dynamics by means of model-theoretic and definability concepts. Definable binding groups precisely capture, in a canonical model-theoretic way, the failure of definable connectedness, the complexity of orbits in definable group actions, and the structure of compactifications of $G$ that are definable in $M$. Their paper underlies the theory of definable topological dynamics, with deep connections to amenability, extreme amenability, and Ellis semigroup theory.

1. Definable Compactification via Model Theory

The universal definable compactification of a group $G$ definable in a structure $M$ is realized as the quotient $G^*/(G^*)_{00}$, where $G^*$ is the interpretation of $G$ in a sufficiently saturated elementary extension $M^*$, and $(G^*)_{00}$ (or sometimes $(G^*)_{00,M}$) denotes the smallest type-definable (over $M$) normal subgroup of $G^*$ of bounded index. The logic topology, in which closed sets are type-definable over $M$, makes $G^*/(G^*)_{00}$ a compact Hausdorff group. A definable compactification of $G$ is a definable homomorphism $h: G \to C$ to a compact group $C$ with dense image, and there is always a universal such compactification, namely $G^* / (G^*)_{00}$ with its naturally induced structure. This construction serves as the model-theoretic analogue of the classical Bohr compactification for definable groups (Gismatullin et al., 2012).

2. Definable G-flows and Universal Ambits

A definable $G$-flow consists of a continuous action of $G$ on a compact Hausdorff space $X$ where, for each $x \in X$, the orbit map $g \mapsto g \cdot x$ is definable in $M$ (i.e., preimages of closed sets under these maps are type-definable in $G$). If all complete types over $M$ concentrating on $G$ are definable, the natural action of $G$ on its type space $S_G(M)$ is definable, and $S_G(M)$ (endowed with the Stone topology) becomes the universal definable $G$-ambit. That is, every definable $G$-ambit factors uniquely and continuously through $S_G(M)$, with the distinguished point corresponding to the type of the identity element (Gismatullin et al., 2012).

3. Minimal Flows and the Role of the Semigroup Structure

Within $S_G(M)$, minimal subflows (closed, $G$-invariant, minimal as such) exist by compactness and correspond to universal minimal definable $G$-flows. The semigroup operation $*$ on $S_G(M)$, given by $p * q = \mathrm{tp}(a b / M)$ for realizations $a \models p$ and $b \models q$ in a monster model, structures $S_G(M)$ as a left topological semigroup. This structure ensures the existence and uniqueness (up to $G$-space isomorphism) of universal minimal definable $G$-flows: minimal left ideals in the semigroup correspond to minimal subflows of $S_G(M)$ and any $G$-map between minimal subflows is an isomorphism (Gismatullin et al., 2012).

4. Amenability, Extreme Amenability, and Genericity

Amenability in the definable context is formulated as the existence of a $G$-invariant Borel probability measure on any definable $G$-flow. Extreme amenability requires every definable $G$-flow to have a fixed point, equivalent to $S_G(M)$ admitting a $G$-invariant type. The latter can be characterized combinatorially: $G$ is definably extremely amenable iff every definable left-generic subset $Y \subseteq G$ (one whose finitely many left translates cover $G$) satisfies $Y \cdot Y^{-1} = G$. This characterizes definable extreme amenability using structural properties of definable sets in $G$ (Gismatullin et al., 2012).

5. Model-Theoretic Perspective on Topological Groups

For a classical topological group $G$ seen as a structure $M$ (with additional predicates for open subsets), the universal (Bohr) compactification and the universal $G$-ambit can also be described using model-theoretic notions: in a sufficiently saturated $M^*$, the intersection of all interpretations $U^*$ of open neighborhoods $U$ of the identity gives a type-definable subgroup $N$ such that $G^*/N$ (with the logic topology) is compact. Similarly, the universal $G$-ambit is obtained as a quotient of $G^*$ by the finest bounded, type-definable over $M$ equivalence relation compatible with the right uniformity. This shows model-theoretic compactification generalizes and encompasses the fundamental constructions of topological group theory (Gismatullin et al., 2012).

6. Integrating Model-Theoretic and Topological Dynamical Notions

The model-theoretic machinery provides a unified language for both definable and classical topological dynamics: compactifications, actions, flows, and the connection with amenability are all encoded via type spaces, definable maps, and type-definable subgroups. The passage from $G$ to $G^*/(G^*)_{00}$ canonically captures the "maximal definable compactification," while $S_G(M)$ and its minimal subflows reflect the universality properties of (minimal) dynamical systems in the definable category. Amenability and extreme amenability become expressible in terms of properties of types and definable subsets, underpinning the translation of classical group-theoretic dynamical information into the model-theoretic context.

7. Summary Table of Central Objects

Model-Theoretic Notion	Topological Analogue	Description
$G^*/(G^*)_{00}$	Bohr compactification	Universal definable compactification
$S_G(M)$	Universal $G$-ambit (Samuel compact)	Stone space of types, universal flow
Minimal subflow of $S_G(M)$	Universal minimal flow	Unique up to $G$-space isomorphism
Definably amenable	Amenable group	Invariant Borel probability measures
Definably extremely amenable	Extremely amenable group	Fixed points in all $G$-flows

Each concept and quotient reflects, when interpreted through the lens of definability and types, a canonical and rigid model-theoretic object mirroring its topological counterpart. The machinery thus provides precise tools to analyze, compactify, and measure the size and complexity of the "binding group" that encapsulates the definable dynamics of $G$.

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The model-theoretic theory of definable binding groups, as detailed in (Gismatullin et al., 2012), thereby offers a comprehensive approach to understanding compactifications, flows, minimality, and invariant measures for groups definable in first-order theories, highlighting the interaction between algebra, topology, and logic, and positioning model theory as a natural setting for generalized topological dynamics.