Related papers: Polish group actions and computability

It is well known that if $G$ is a countable amenable group and $G \curvearrowright (Y, \nu)$ factors onto $G \curvearrowright (X, \mu)$, then the entropy of the first action must be greater than or equal to the entropy of the second action.…

The motivation of this article is to introduce a kind of orbit equivalence relations which can well describe structures and properties of Polish groups from the perspective of Borel reducibility. Given a Polish group $G$, let $E(G)$ be the…

We study when a continuous isometric action of a Polish group on a complete metric space is, or can be, transitive. Our main results consist of showing that certain Polish groups, namely $\mathrm{Aut}^*(\mu)$ and $\mathrm{Homeo}^+[0,1]$,…

Let K be an expansion of either an ordered field or a valued field. Given a definable set X $\subseteq$ K^m let C(X) be the ring of continuous definable functions from X to K. Under very mild assumptions on the geometry of X and…

We prove some technical results on definable types in $p$-adically closed fields, with consequences for definable groups and definable topological spaces. First, the code of a definable $n$-type (in the field sort) can be taken to be a real…

We use Fra\" iss\'e theoretic methods to construct several universal and ultrahomogeneous Polish metric structures. Namely, universal and ultrahomogeneous Polish metric space equipped with countably many closed subsets of its powers,…

Let X be an infinite set of regular cardinality. We determine all clones on X which contain all almost unary functions. It turns out that independently of the size of X, these clones form a countably infinite descending chain. Moreover, all…

We initiate the study of computable presentations of real and complex C*-algebras under the program of effective metric structure theory. With the group situation as a model, we develop corresponding notions of recursive presentations and…

We consider clones on countable sets. If such a clone has quasigroup operations, is locally closed and countable, then there is a function $f : \mathbb{N} \to \mathbb{N}$ such that the $n$-ary part of $C$ is equal to the $n$-ary part of…

We prove that for each Polish space X, the space C(X) of continuous real-valued functions on X satisfies a strong version of the Pytkeev property, if endowed with the compact-open topology. (This shows that whereas it need not be…

Given Polish space ${\bf Y}$ and continuous language $L$ we study the corresponding logic $\mathsf{Iso}({\bf Y})$-space ${\bf Y}_L$. We build a framework of generalized model theory towards analysis of Borel/algorithmic complexity of…

If G is a locally essential subgroup of a compact abelian group K, then: (i) t(G)=w(G)=w(K), where t(G) is the tightness of G; (ii) if G is radial, then K must be metrizable; (iii) G contains a super-sequence S converging to 0 such that…

We consider factorizations of a finite group $G$ into conjugate subgroups, $G=A^{x_{1}}\cdots A^{x_{k}}$ for $A\leq G$ and $x_{1},\ldots ,x_{k}\in G$, where $A$ is nilpotent or solvable. First we exploit the split $BN$-pair structure of…

We prove that no uncountable Polish group can admit a system of generators whose associated length function satisfies the following conditions: (i) if $0 < k < \omega$, then $lg(x) \leq lg(x^k)$; (ii) if $lg(y) < k < \omega$ and $x^k = y$,…

A Polish group $G$ has the generic point property if any minimal $G$-flow admits a comeager orbit, or equivalently if the universal minimal flow (UMF) does. The class $\mathsf{GPP}$ of such Polish groups is a proper extension of the class…

Answering a question of Gao and Kechris, we show that, given any polish group G, there exists a closed subset F of Urysohn's universal metric space U such that G is (topologically) isomorphic to the subgroup of isometries of U which map F…

In this paper we settle most of the open questions on algorithmic computability of Julia sets. In particular, we present an algorithm for constructing quadratics whose Julia sets are uncomputable. We also show that a filled Julia set of a…

In recent years, much work has been done to measure and compare the complexity of orbit equivalence relations, especially for certain classes of Polish groups. We start by introducing some language to organize this previous work, namely the…

It is consistent that there is a partial order (P,<) of size aleph_1 such that every monotone (unary) function from P to P is first order definable in (P,<). The partial order is constructed in an extension obtained by finite support…

Suppose that the finite group $G=AB$ is a mutually permutable product of two subgroups $A$ and $B$. By using Sylow numbers of $A$ and $B$, we present some new bounds of the $p$-length $l_p(G)$ of a $p$-solvable group $G$ and the nilpotent…