Ellis Group Research Articles

Ramsey theory and topological dynamics for first order theories

We investigate interactions between Ramsey theory, topological dynamics, and model theory. We introduce various Ramsey-like properties for first order theories and characterize them in terms of the appropriate dynamical properties of the theories in question (such as [extreme] amenability of a theory or some properties of the associated Ellis semigroups). Then we relate them to profiniteness and triviality of the Ellis groups of first order theories. In particular, we find various criteria for [pro]finiteness and for triviality of the Ellis group of a given theory from which we obtain wide classes of examples of theories with [pro]finite or trivial Ellis groups. We also find several concrete examples illustrating the lack of implications between some fundamental properties. In the appendix, we give a full computation of the Ellis group of the theory of the random hypergraph with one binary and one 4-ary relation. This example shows that the assumption of NIP in the version of Newelski’s conjecture for amenable theories (proved by Krupiński, Newelski, and Simon [J. Math. Log. 19 (2019), no. 2, 1950012, p. 55]) cannot be dropped.

Definably topological dynamics of p-adic algebraic groups

We study the p-adic algebraic groups G from the definable topological-dynamical point of view. We consider the case where M is an arbitrary p-adic closed field and G an algebraic group over Qp admitting an Iwasawa decompostion G=KB, where K is open and definably compact over Qp, and B is a Borel subgroup of G over Qp. Our main result is an explicit description of the minimal subflow and Ellis group of the universal definable G(M)-flow SG(Mext). We prove that the Ellis group of SG(Mext) is isomorphic to the Ellis group of SB(Mext), which is B/B0.As applications, we conclude that the Ellis groups corresponding to GL(n,M) and SL(n,M) are isomorphic to (Zˆ×Zp⁎)n and (Zˆ×Zp⁎)n−1 respectively, generalizing the main result of Penazzi, Pillay, and Yao in [21].

Topological dynamics and NIP fields

We study definable topological dynamics of some algebraic group actions over an arbitrary NIP field K. We show that the Ellis group of the universal definable flow of SL2(K) is non-trivial if the multiplicative group of K is not type-definably connected, providing a way to find multiple counterexamples to the (model-theoretic) Ellis group conjecture, particularly in the case of dp-minimal fields. We also study some structure theory of algebraic groups over K with definable f-generics.

Boundedness and absoluteness of some dynamical invariants in model theory

Let [Formula: see text] be a monster model of an arbitrary theory [Formula: see text], let [Formula: see text] be any (possibly infinite) tuple of bounded length of elements of [Formula: see text], and let [Formula: see text] be an enumeration of all elements of [Formula: see text] (so a tuple of unbounded length). By [Formula: see text] we denote the compact space of all complete types over [Formula: see text] extending [Formula: see text], and [Formula: see text] is defined analogously. Then [Formula: see text] and [Formula: see text] are naturally [Formula: see text]-flows (even [Formula: see text]-ambits). We show that the Ellis groups of both these flows are of bounded size (i.e. smaller than the degree of saturation of [Formula: see text]), providing an explicit bound on this size. Next, we prove that these Ellis groups do not depend (as groups equipped with the so-called [Formula: see text]-topology) on the choice of the monster model [Formula: see text]; thus, we say that these Ellis groups are absolute. We also study minimal left ideals (equivalently subflows) of the Ellis semigroups of the flows [Formula: see text] and [Formula: see text]. We give an example of a NIP theory in which the minimal left ideals are of unbounded size. Then we show that in each of these two cases, boundedness of a minimal left ideal (equivalently, of all the minimal left ideals) is an absolute property (i.e. it does not depend on the choice of [Formula: see text]) and that whenever such an ideal is bounded, then in some sense its isomorphism type is also absolute. Under the assumption that [Formula: see text] has NIP, we give characterizations (in various terms) of when a minimal left ideal of the Ellis semigroup of [Formula: see text] is bounded. Then we adapt the proof of Theorem 5.7 in Definably amenable NIP groups, J. Amer. Math. Soc. 31 (2018) 609–641 to show that whenever such an ideal is bounded, a certain natural epimorphism (described in [K. Krupiński, A. Pillay and T. Rzepecki, Topological dynamics and the complexity of strong types, Israel J. Math. 228 (2018) 863–932]) from the Ellis group of the flow [Formula: see text] to the Kim–Pillay Galois group [Formula: see text] is an isomorphism (in particular, [Formula: see text] is G-compact). We also obtain some variants of these results, formulate some questions, and explain differences (providing a few counterexamples) which occur when the flow [Formula: see text] is replaced by [Formula: see text].

On the Ellis Group of a Class of Distal Groups and a Problem by Milnes

In (Acta Math Hangar 65(2):149–164, 1994), Milnes raised the problem that whether the Ellis group of a distal group is itself distal. In this paper, we give a positive answer to Milnes problem for the case of group extensions of Abelian groups. We also investigate the Ellis groups of a well-known family of distal compact right topological groups, namely the groups $$ E\left( {\mathbb{T}} \right)^{\infty } , E\left( {\mathbb{T}} \right)^{k} $$ and $$ E\left( {\mathbb{T}} \right)^{k - 1} \times {\mathbb{T}} $$, where $$ E\left( {\mathbb{T}} \right) $$ is the group of all endomorphisms on the unit circle $$ {\mathbb{T}} $$.

Some model theory and topological dynamics of $p$-adic algebraic groups

We initiate the study of p-adic algebraic groups G from the stability-theoretic and definable topological-dynamical points of view, that is, we consider invariants of the action of G on its space of types over Q_p in the language of fields. We consider the additive and multiplicative groups of Q_p and Z_p, the group of upper triangular invertible 2\times 2 matrices, SL(2,Z_p), and, our main focus, SL(2,Q_p). In all cases we identify f-generic types (when they exist), minimal subflows, and idempotents. Among the main results is that the ``Ellis group" of SL(2,Q_p)$ is the profinite completion of Z, yielding a counterexample to Newelski's conjecture with new features: G = G^{00} = G^{000} but the Ellis group is infinite. A final section deals with the action of SL(2,Q_p) on the type-space of the projective line over Q_p.

THE ELLIS GROUP CONJECTURE AND VARIANTS OF DEFINABLE AMENABILITY

AbstractWe consider definable topological dynamics for NIP groups admitting certain decompositions in terms of specific classes of definably amenable groups. For such a group, we find a description of the Ellis group of its universal definable flow. This description shows that the Ellis group is of bounded size. Under additional assumptions, it is shown to be independent of the model, proving a conjecture proposed by Newelski. Finally we apply the results to new classes of groups definable in o-minimal structures, generalizing all of the previous results for this setting.

Furstenberg–Ellis–Namioka Structure Theorem on a CHART Group

For $$E({\mathbb {T}})$$ being the endomorphism group of the circle group $${\mathbb {T}}$$ , the Furstenberg–Ellis–Namioka Structure Theorem of the CHART group $$G=E({\mathbb {T}})\times {\mathbb {T}}$$ with the product $$(f,u)(g,v)=(fg,uvf\circ g(\mathrm{e}^{i}))$$ is known to be equal to $$\{G,1_{\mathbb {T}}\times {\mathbb {T}},\{(1_{\mathbb {T}},1)\}\}$$ . A somewhat similar group structure is known to exist on $$E({\mathbb {T}})\times E({\mathbb {T}})\times {\mathbb {T}}$$ , studied by Milnes. We give an explicit characterization of the Furstenberg–Ellis–Namioka Structure Theorem for an admissible subgroup $$\Sigma $$ of $$E({\mathbb {T}})\times E({\mathbb {T}})\times {\mathbb {T}}$$ , where $$\Sigma $$ is the Ellis group of the Hahn-type skew product dynamical system on the 3-torus $${\mathbb {T}}^3$$ .

Definably amenable NIP groups

We study definably amenable NIP groups. We develop a theory of generics showing that various definitions considered previously coincide, and we study invariant measures. As applications, we characterize ergodic measures, give a proof of the conjecture of Petrykowski connecting existence of bounded orbits with definable amenability in the NIP case, and prove the Ellis group conjecture of Newelski and Pillay connecting the model-theoretic connected component of an NIP group with the ideal subgroup of its Ellis enveloping semigroup.

DEFINABLE TOPOLOGICAL DYNAMICS

AbstractFor a group G definable in a first order structure M we develop basic topological dynamics in the category of definable G-flows. In particular, we give a description of the universal definable G-ambit and of the semigroup operation on it. We find a natural epimorphism from the Ellis group of this flow to the definable Bohr compactification of G, that is to the quotient ${G^{\rm{*}}}/G_M^{{\rm{*}}00}$ (where G* is the interpretation of G in a monster model). More generally, we obtain these results locally, i.e., in the category of Δ-definable G-flows for any fixed set Δ of formulas of an appropriate form. In particular, we define local connected components $G_{{\rm{\Delta }},M}^{{\rm{*}}00}$ and $G_{{\rm{\Delta }},M}^{{\rm{*}}000}$, and show that ${G^{\rm{*}}}/G_{{\rm{\Delta }},M}^{{\rm{*}}00}$ is the Δ-definable Bohr compactification of G. We also note that some deeper arguments from [14] can be adapted to our context, showing for example that our epimorphism from the Ellis group to the Δ-definable Bohr compactification factors naturally yielding a continuous epimorphism from the Δ-definable generalized Bohr compactification to the Δ-definable Bohr compactification of G. Finally, we propose to view certain topological-dynamic and model-theoretic invariants as Polish structures which leads to some observations and questions.

Definable topological dynamics and real Lie groups

We investigate definable topological dynamics of groups definable in an o‐minimal expansion of the field of reals. Assuming that a definable group G admits a model‐theoretic analogue of Iwasawa decomposition, namely the compact‐torsion‐free decomposition , we give a description of minimal subflows and the Ellis group of its universal definable flow in terms of this decomposition. In particular, the Ellis group of this flow is isomorphic to . This provides a range of counterexamples to a question by Newelski whether the Ellis group is isomorphic to . We further extend the results to universal topological covers of definable groups, interpreted in a two‐sorted structure containing the o‐minimal sort and a sort for an abelian group.

Topological dynamics for groups definable in real closed field

We study the definable topological dynamics of groups definable in an o-minimal expansion of an arbitrary real closed field M. For a definable group G which admits a compact-torsion-free decomposition G=HK, we give a description of the minimal subflow and Ellis group of the universal definable G(M)-flow SG,ext(M). This Ellis group is isomorphic to NG(H)∩K(R), which extends the result of G. Jagiella from [7]. We also consider SL(2,M) as an example, explaining the difference between the universal definable SL(2,R)-flow, SG(R) and the universal definable G(M)-flow, SG,ext(M) for an arbitrary model M≻R.

External definability and groups in NIP theories

We prove that many properties and invariants of definable groups in NIP theories (i.e. theories without the independence property), such as definable amenability, G / G 00 , etc., are preserved when passing to the theory of the Shelah expansion by externally definable sets, M ext , of a model M. In the light of these results, we continue the study of the ‘definable topological dynamics’ of groups in NIP theories. In particular, we prove the Ellis group conjecture relating the Ellis group to G / G 00 in some new cases, including definably amenable groups in o-minimal structures.

Toward Position-independent Retroviral Vector Expression in Pluripotent Stem Cells

Toward Position-independent Retroviral Vector Expression in Pluripotent Stem Cells

Evolution of the Cordilleran foreland basin system in northwestern Montana, U.S.A.

New lithostratigraphic and chronostratigraphic, geochronologic, and sedimentary petrologic data illuminate the history of development of the North American Cordilleran foreland basin system and adjacent thrust belt from Middle Jurassic through Eocene time in northwestern Montana. The oldest deposits in the foreland basin system consist of relatively thin, regionally tabular deposits of the marine Ellis Group and fluvial-estuarine Morrison Formation, which accumulated during Bajocian to Kimmeridgian time. U-Pb ages of detrital zircons and sandstone modal petrographic data indicate that by ca. 170 Ma, miogeoclinal strata were being deformed and eroded in hinterland regions. Sandstones of the Swift and Morrison Formations contain detrital zircons derived from the Intermontane belt. The Jurassic deposits probably accumulated in the distal, back-bulge depozone of an early foreland basin, as suggested by the slow rates of tectonic subsidence and tabular geometry. A regional unconformity separates the Jurassic strata from late Barremian(?) foredeep deposits. This unconformity possibly resulted as a combined effect of forebulge migration, decreased dynamic subsidence, and eustatic sea-level fall. The late Barremian(?)-early Albian Kootenai Formation is the first unit that consistently thickens westward, as would be expected in a foredeep depozone. The subsidence curve at this time begins to show the convex-upward pattern characteristic of foredeeps. By Albian time, the fold-and-thrust belt had propagated to the east and incorporated Proterozoic rocks of the Belt Supergroup, as indicated by sandstone compositions, detrital zircon ages in the Blackleaf Formation, and by crosscutting relationships in thrust sheets involving Belt Supergroup rocks in the thrust belt. A major episode of marine inundation and black shale deposition (Marias River Shale) occurred between the Cenomanian and mid-Santonian, and was followed by a regressive succession represented by the Upper Santonian–mid-Campanian Telegraph Creek, Virgelle, and Two Medicine Formations. Provenance data do not resolve the timing of individual thrust displacements during Cenomanian–early Campanian time. The Upper Campanian Bearpaw Formation represents the last major marine inundation in the foreland basin. By latest Campanian time, a major episode of slip on the Lewis thrust system had commenced, as recorded in the foreland by the Willow Creek and St. Mary River Formations in the proximal foredeep depozone. The final stage in the evolution of the Cordilleran fold-and-thrust belt and foreland basin system is recorded by the Paleocene–early Eocene Fort Union and Wasatch Formations, which were preserved in the distal foreland region. Regional extensional faulting along the fold-and-thrust belt began during the middle Eocene. The results presented here enable the establishment of links between previous geological work in Canada and the better known parts of the Cordilleran foreland basin in the United States.

The topological center of the spectrum of some distal algebras

The topological center of the spectrum of the Weyl algebra W, i.e. the norm closure of the algebra generated by the set of functions $\{n\mapsto\lambda^{n^{i}};\lambda\in\mathbb{T}\mbox{ and }i\in\mathbb {N}\}$, is characterized in a recent paper by Jabbari and Namioka (Ellis group and the topological center of the flow generated by the map $n\mapsto \lambda^{n^{k}}$, to appear in Milan J. Math.). By the techniques essentially used in the cited paper, the topological center of the spectrum of the subalgebra Wk, the norm closure of the algebra generated by the set of functions $\{n\mapsto\lambda^{n^{i}};\lambda\in\mathbb{T}\mbox{ and }i=0,1,2,\ldots,k\}$, will be characterized, for all k∈ℕ. Also an example of a non-minimal dynamical system, with the enveloping semigroup Σ, for which the set of all continuous elements of Σ is not equal to the topological center of Σ, is given.

Ellis Group and the Topological Center of the Flow Generated by the Map $${n \mapsto {\lambda^{n}}^{k}}$$

For each natural number k and each irrational member λ of the unit circle, it is proved that the shift-orbit closure Xf of the function \({f(n) = {\lambda^{n}}^{k}}\) is homeomorphic to a k-torus. Using this homeomorphism, we investigate the Ellis group and its topological center of the dynamical system (Xf, U), where U is the shift operator on \({l^{\infty}(\mathbb{Z})}\). Finally, it is shown that the topological center of the spectrum of the Weyl algebra is the image of \({\mathbb{Z}}\) in the spectrum.

Jurassic onset of foreland basin deposition in northwestern Montana, USA: Implications for along-strike synchroneity of Cordilleran orogenic activity

Stratigraphic, provenance, and subsidence analyses suggest that by the Middle to Late Jurassic a foreland basin system was active in northwestern Montana (United States). U-Pb ages of detrital zircons and detrital modes of sandstones indicate provenance from accreted terranes and deformed miogeoclinal rocks to the west. Subsidence commenced ca. 170 Ma and followed a sigmoidal pattern characteristic of foreland basin systems. Thin Jurassic deposits of the Ellis Group and Morrison Formation accumulated in a backbulge depozone. A regional unconformity and/or paleosol zone separates the Morrison from Early Cretaceous foredeep deposits of the Kootenai Formation. The model presented here is consistent with regional deformation events registered in hinterland regions, and challenges previous interpretations of a strongly diachronous onset of Cordilleran foreland basin deposition from northwestern Montana to southern Canada.

Mixed sediment deposition in a retro-arc foreland basin: Lower Ellis Group (M. Jurassic), Wyoming and Montana, U.S.A.

The “lower” Ellis Group (M. Jurassic) of northern Wyoming and southern Montana affords an excellent opportunity to examine the influence of tectonics, sea-level change, and incipient topography on facies dynamics and the evolution of mixed sediment ramp deposits. The Sawtooth, Piper, and Gypsum Spring formations (Bajocian to Callovian) represent sedimentation along the forebulge of a retro-arc foreland basin. The “lower” Ellis Group records deposition during two transgressive–regressive cycles, (1) a Bajocian-age cycle dominated by evaporites and red shales, and (2) a Bathonian-age cycle characterized by carbonates, evaporites, and red shales. These cycles are capped by a Callovian-age cycle distinguished by carbonates and red shales that is represented by the “lower” Sundance and Rierdon Formations. Transgressive episodes favored intensified chemical sediment production resulting in thick units deposited in subtidal to peritidal environments. Regressive periods are characterized by supratidal redbed progradation and subsequent shallowing–upward cycles. The depositional cycles in the lower Ellis Group developed due the interplay between sea-level change and tectonic subsidence related to the evolution of a retro-arc foreland basin. Differential subsidence before, during, and after deposition created paleohighs that locally influenced accommodation space and, thereby, complicated depositional and erosional patterns. This paper provides a regional framework for further analysis of the depositional history of the lower Ellis Group by addressing the stratigraphic relationships between the Sawtooth, Piper, and Gypsum Spring Formations.

Sedimentation in a low-accommodation setting: Nonmarine (Cretaceous) Mannville and marine (Jurassic) Ellis groups, Manyberries field, southeastern Alberta

Nonmarine strata of the Lower Cretaceous Mannville Group and marine strata of the Upper Jurassic Ellis Group in southern Alberta were deposited in the low-accommodation, eastern, and distal margin of the Western Canada sedimentary basin. The Manyberries oil field produces hydrocarbons from preserved interfluves as well as the deposits of compound valley fills. These Mannville Group sediments provide an end-member component for nonmarine sedimentation in a low-accommodation setting. The succession is characterized by thin sediment accumulations deposited over a long period of time; multiple unconformities, some profound; incised valleys, typically compound in nature; paleosols of varying degrees and maturity; and a general absence of coal.