Measure conjugacy invariants for actions of countable sofic groups

We introduce topological pressure for continuous actions of countable sofic groups on compact metrizable spaces. This generalizes the classical topological pressure for continuous actions of countable amenable groups on such spaces. We also establish the variational principle for topological pressure in this sofic context.

We prove that, aside from the obvious exceptions, the mapping class group of a compact orientable surface is not abstractly commensurable with any right-angled Artin group. Our argument applies to various sub- groups of the mapping class group—the subgroups generated by powers of Dehn twists and the terms of the Johnson filtration—and additionally to the outer automorphism group of a free group and to certain linear groups. There are many analogies and interconnections between the theories of right- angled Artin groups on one hand and mapping class groups on the other hand. For instance, by the work of Crisp and Wiest (13), the work of Koberda (27), and the work of the first two authors with Mangahas (12), there is an abundance of injective homomorphisms from right-angled Artin groups to mapping class groups. Also, the last two authors proved (30) that any two elements of the pure braid group either generate a free group or a free abelian group—a property shared by all right-angled Artin groups (4, Theorem 1.2). We are thus led to ask to what extent mapping class groups are the same as right-angled Artin groups. It is straightforward to see that most mapping class groups are not isomorphic to right-angled Artin groups, for instance because right-angled Artin groups are torsion free. On the other hand, mapping class groups have finite-index sub- groups that are torsion free, and so this leaves open the possibility that mapping class groups are abstractly commensurable to right-angled Artin groups, that is, that they have isomorphic finite-index subgroups. We prove that, aside from a small number of exceptions, this is not the case. We also extend this result to several classes of groups related to mapping class groups. We start by recalling some definitions. To a finite graph , we can associate a right-angled Artin grou p: this is the group with one generator for each vertex of , and one defin ing relator for each edge, namely, the commutator of the two generators corresponding to the endpoints. Let Sg,n denote a closed, connected, orientable surface of genus g with n marked points. The mapping class group Mod(Sg,n) is the group of homotopy classes of orientation-preserving homeomorphisms of Sg,n preserving the set of marked points. As discussed in Koberda's paper (27, Theorem 1.5), no finite-index subgroup of Mod(Sg,n) injects into a right-angled Artin group if g ≥ 2 and (g,n) 6 (2,0); see

<p style='text-indent:20px;'>In this article we study the centralizer of a minimal aperiodic action of a countable group on the Cantor set (an aperiodic minimal Cantor system). We show that any countable residually finite group is the subgroup of the centralizer of some minimal <inline-formula><tex-math id="M1">\begin{document}$ \mathbb Z $\end{document}</tex-math></inline-formula> action on the Cantor set, and that any countable group is the subgroup of the normalizer of a minimal aperiodic action of an abelian countable free group on the Cantor set. On the other hand we show that for any countable group <inline-formula><tex-math id="M2">\begin{document}$ G $\end{document}</tex-math></inline-formula>, the centralizer of any minimal aperiodic <inline-formula><tex-math id="M3">\begin{document}$ G $\end{document}</tex-math></inline-formula>-action on the Cantor set is a subgroup of the centralizer of a minimal <inline-formula><tex-math id="M4">\begin{document}$ \mathbb Z $\end{document}</tex-math></inline-formula>-action.

In a paper entitled cocycles and spectra Helson and Par ry prove that for every T there exists a q~ such that V ~ has Lebesgue spectrum, moreover q~ can be chosen to be real, i.e., taking values + 1 and 1. The purpose of this paper is to extend this result to certain actions of countable groups which includes ergodic non-singular actions of countable abelian groups. We blend the method of Helson and Parry with the notions of weak equivalence and weak von Neumann transformations. In section 3 and 4 we discuss these results in connection with systems of imprimitivity. The problem of extending the result quoted above to countable groups was raised by H. Helson to one of us. It is a pleasure to acknowledge his interest and encouragement in this work.

Let $T$ be a measure-preserving and mixing action of a countable abelian group $G$ on a probability space $(X,\mathscr S,\mu )$ and $A$ a locally compact second countable abelian group. A cocycle $c\colon G\times X\to A$ for $T$ <i>disperses</i> if $\lim_

We show that any pivotal Hopf monoid H in a symmetric monoidal category \mathcal{C} gives rise to actions of mapping class groups of oriented surfaces of genus g\geq 1 with n\geq 1 boundary components. These mapping class group actions are given by group homomorphisms into the group of automorphisms of certain Yetter–Drinfeld modules over H . They are associated with edge slides in embedded ribbon graphs that generalise chord slides in chord diagrams. We give a concrete description of these mapping class group actions in terms of generating Dehn twists and defining relations. For the case where \mathcal{C} is finitely complete and cocomplete, we also obtain actions of mapping class groups of closed surfaces by imposing invariance and coinvariance under the Yetter–Drinfeld module structure.

We consider finite-sheeted, regular, possibly branched covering spaces of compact surfaces with boundary and the associated liftable and symmetric mapping class groups. In particular, we classify when either of these subgroups coincides with the entire mapping class group of the surface. As a consequence, we construct infinite families of non-geometric embeddings of the braid group into mapping class groups in the sense of Wajnryb. Indeed, our embeddings map standard braid generators to products of Dehn twists about curves forming chains of arbitrary length. As key tools, we use the Birman-Hilden theorem and the action of the mapping class group on a particular fundamental groupoid of the surface.

In this paper we consider a bootstrap class $$\mathfrak {C}$$ of countable discrete groups, which is closed under countable unions and extensions by the integers, and we study actions of such groups on $$\mathrm {C}^*$$ -algebras. This class includes all torsion-free abelian groups, poly- $$\mathbb {Z}$$ -groups, as well as other examples. Using the interplay between relative Rokhlin dimension and semi-strongly self-absorbing actions established in prior work, we obtain the following two main results for any group $$\Gamma \in \mathfrak {C}$$ and any strongly self-absorbing $$\mathrm {C}^*$$ -algebra $$\mathcal {D}$$ : In fact we establish more general relative versions of these two results for actions of amenable groups that have a predetermined quotient in the class $$\mathfrak {C}$$ . For the monotracial case, the proof comprises an application of Matui–Sato’s equivariant property (SI) as a key method.

Let Σ be an orientable surface of finite type and let Mod(Σ) be its mapping class group. We consider actions of Mod(Σ) by semisimple isometries on complete CAT(0) spaces. If the genus of Σ is at least 3, then in any such action all Dehn twists act as elliptic isometries. The action of Mod(Σ) on the completion of Teichmüller space with the Weil-Petersson metric shows that there are interesting actions of this type. Whenever the mapping class group of a closed orientable surface of genus g acts by semisimple isometries on a complete CAT(0) space of dimension less than g it must fix a point. The mapping class group of a closed surface of genus 2 acts properly by semisimple isometries on a complete CAT(0) space of dimension 18.

Abstract

This book contains a collection of survey papers by leading researchers in ergodic theory, low-dimensional and topological dynamics and it comprises nine chapters on a range of important topics. These include: the role and usefulness of ultrafilters in ergodic theory, topological dynamics and Ramsey theory; topological aspects of kneading theory together with an analogous 2-dimensional theory called pruning; the dynamics of Markov odometers, Bratteli-Vershik diagrams and orbit equivalence of non-singular automorphisms; geometric proofs of Mather's connecting and accelerating theorems; recent results in one dimensional smooth dynamics; periodic points of nonexpansive maps; arithmetic dynamics; the defect of factor maps; entropy theory for actions of countable amenable groups.

In this note, we prove that if G is a countable group that contains a nonabelian free subgroup then every pair of nontrivial Bernoulli shifts over G are weakly isomorphic.

We introduce a technique for proving quantitative representation stability theorems for sequences of representations of certain finite linear groups over a field of characteristic zero. In particular, we prove a vanishing result for higher syzygies of VIC- and SI-modules, which can be thought of as a weaker version of a regularity theorem of Church-Ellenberg in the context of FI-modules. We apply these techniques to the rational homology of congruence subgroups of mapping class groups and congruence subgroups of automorphism groups of free groups. This partially resolves a question raised by Church and Putman-Sam. We also prove new homological stability results for mapping class groups and automorphism groups of free groups with twisted coefficients.

McCarthy's Theorem for the mapping class group of a closed hyperbolic surface states that for any two mapping classes $\sigma,\tau \in \mathrm{Mod}(S)$ there is some power $N$ such that the group $\langle \sigma^N,\tau^N\rangle$ is either free of rank two or abelian, and gives a geometric criterion for the dichotomy. The analogous statement is false in linear groups, and unresolved for outer automorphisms of a free group. Several analogs are known for exponentially growing outer automorphisms satisfying various technical hypothesis. In this article we prove an analogous statement when $\sigma$ and $\tau$ are linearly growing outer automorphisms of $F_r$, and give a geometric criterion for the dichotomy. Further, Hamidi-Tehrani proved that for Dehn twists in the mapping class group this independence dichotomy is \emph{uniform}: $N=4$ suffices. In a similar style, we obtain an $N$ that depends only on the rank of the free group.

Cycle indices of linear, affine, and projective groups