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Register a new userBoundedness and absoluteness of some dynamical invariants in model theory
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Let ${mathfrak C}$ be a monster model of an arbitrary theory $T$, $bar alpha$ any tuple of bounded length of elements of ${mathfrak C}$, and $bar c$ an enumeration of all elements of ${mathfrak C}$. By $S_{bar alpha}({mathfrak C})$ denote the compact space of all complete types over ${mathfrak C}$ extending $tp(bar alpha/emptyset)$, and $S_{bar c}({mathfrak C})$ is defined analogously. Then $S_{bar alpha}({mathfrak C})$ and $S_{bar c}({mathfrak C})$ are naturally $Aut({mathfrak C})$-flows. We show that the Ellis groups of both these flows are of bounded size (i.e. smaller than the degree of saturation of ${mathfrak C}$), providing an explicit bound on this size. Next, we prove that these Ellis groups do not depend on the choice of the monster model ${mathfrak C}$; thus, we say that they are absolute. We also study minimal left ideals (equivalently subflows) of the Ellis semigroups of the flows $S_{bar alpha}({mathfrak C})$ and $S_{bar c}({mathfrak C})$. We give an example of a NIP theory in which the minimal left ideals are of unbounded size. We show that in each of these two cases, boundedness of a minimal left ideal is an absolute property (i.e. it does not depend on the choice of ${mathfrak C}$) and that whenever such an ideal is bounded, then its isomorphism type is also absolute. Assuming NIP, we give characterizations of when a minimal left ideal of the Ellis semigroup of $S_{bar c}({mathfrak C})$ is bounded. Then we adapt a proof of Chernikov and Simon to show that whenever such an ideal is bounded, the natural epimorphism (described by Krupinski, Pillay and Rzepecki) from the Ellis group of the flow $S_{bar c}({mathfrak C})$ to the Kim-Pillay Galois group $Gal_{KP}(T)$ is an isomorphism (in particular, $T$ is G-compact). We provide some counter-examples for $S_{bar alpha}({mathfrak C})$ in place of $S_{bar c}({mathfrak C})$.
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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 2times 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 Newelskis 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.
In [FHK13], the authors considered the question whether model-existence of $L_{omega_1,omega}$-sentences is absolute for transitive models of ZFC, in the sense that if $V subseteq W$ are transitive models of ZFC with the same ordinals, $varphiin V$ and $Vmodels varphi text{ is an } L_{omega_1,omega}text{-sentence}$, then $V models varphi text{ has a model of size } aleph_alpha$ if and only if $W models varphi text{ has a model of size } aleph_alpha$. From [FHK13] we know that the answer is positive for $alpha=0,1$ and under the negation of CH, the answer is negative for all $alpha>1$. Under GCH, and assuming the consistency of a supercompact cardinal, the answer remains negative for each $alpha>1$, except the case when $alpha=omega$ which is an open question in [FHK13]. We answer the open question by providing a negative answer under GCH even for $alpha=omega$. Our examples are incomplete sentences. In fact, the same sentences can be used to prove a negative answer under GCH for all $alpha>1$ assuming the consistency of a Mahlo cardinal. Thus, the large cardinal assumption is relaxed from a supercompact in [FHK13] to a Mahlo cardinal. Finally, we consider the absoluteness question for the $aleph_alpha$-amalgamation property of $L_{omega_1,omega}$-sentences (under substructure). We prove that assuming GCH, $aleph_alpha$-amalgamation is non-absolute for $1<alpha<omega$. This answers a question from [SS]. The cases $alpha=1$ and $alpha$ infinite remain open. As a corollary we get that it is non-absolute that the amalgamation spectrum of an $L_{omega_1,omega}$-sentence is empty.
We prove several theorems relating amenability of groups in various categories (discrete, definable, topological, automorphism group) to model-theoretic invariants (quotients by connected components, Lascar Galois group, G-compactness, ...). For example, if $M$ is a countable, $omega$-categorical structure and $Aut(M)$ is amenable, as a topological group, then the Lascar Galois group $Gal_{L}(T)$ of the theory $T$ of $M$ is compact, Hausdorff (also over any finite set of parameters), that is $T$ is G-compact. An essentially special case is that if $Aut(M)$ is extremely amenable, then $Gal_{L}(T)$ is trivial, so, by a theorem of Lascar, the theory $T$ can be recovered from its category $Mod(T)$ of models. On the side of definable groups, we prove for example that if $G$ is definable in a model $M$, and $G$ is definably amenable, then the connected components ${G^{*}}^{00}_{M}$ and ${G^{*}}^{000}_{M}$ coincide, answering positively a question from an earlier paper of the authors. We also take the opportunity to further develop the model-theoretic approach to topological dynamics, obtaining for example some new invariants for topological groups, as well as allowing a uniform approach to the theorems above and the various categories.
The flex locus parameterizes plane cubics with three collinear cocritical points under a projection, and the gothic locus arises from quadratic differentials with zeros at a fiber of the projection and with poles at the cocritical points. The flex and gothic loci provide the first example of a primitive, totally geodesic subvariety of moduli space and new ${rm SL}_2(mathbb{R})$-invariant varieties in Teichmuller dynamics, as discovered by McMullen-Mukamel-Wright. In this paper we determine the divisor class of the flex locus as well as various tautological intersection numbers on the gothic locus. For the case of the gothic locus our result confirms numerically a conjecture of Chen-Moller-Sauvaget about computing sums of Lyapunov exponents for ${rm SL}_2(mathbb{R})$-invariant varieties via intersection theory.
In [5], Hjorth proved that for every countable ordinal $alpha$, there exists a complete $mathcal{L}_{omega_1,omega}$-sentence $phi_alpha$ that has models of all cardinalities less than or equal to $aleph_alpha$, but no models of cardinality $aleph_{alpha+1}$. Unfortunately, his solution does not yield a single $mathcal{L}_{omega_1,omega}$-sentence $phi_alpha$, but a set of $mathcal{L}_{omega_1,omega}$-sentences, one of which is guaranteed to work. It was conjectured in [9] that it is independent of the axioms of ZFC which of these sentences has the desired property. In the present paper, we prove that this conjecture is true. More specifically, we isolate a diagonalization principle for functions from $omega_1$ to $omega_1$ which is a consequence of the Bounded Proper Forcing Axiom (BPFA) and then we use this principle to prove that Hjorths solution to characterizing $aleph_2$ in models of BPFA is different than in models of CH. In addition, we show that large cardinals are not needed to obtain this independence result by proving that our diagonalization principle can be forced over models of CH.
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