Free Group Actions Research Articles

The Shannon–McMillan–Breiman theorem beyond amenable groups

We introduce a new isomorphism-invariant notion of entropy for measure-preserving actions of arbitrary countable groups on probability spaces, which we call orbital Rokhlin entropy. It employs Danilenko’s orbital approach to entropy of a partition, and it is motivated by Seward’s recent generalization of Rokhlin’s characterization of entropy from amenable to general groups. A key ingredient in our approach is the use of an auxiliary probability-measure-preserving hyperfinite equivalence relation. Under the assumption of ergodicity of the auxiliary equivalence relation, our main result is a Shannon–McMillan–Breiman pointwise almost sure convergence theorem for the orbital entropy of partitions in measure-preserving group actions, the first such convergence result going beyond the realm of amenable groups. As a special case, we obtain a Shannon–McMillan–Breiman theorem for all strongly mixing actions of any countable group. Furthermore, we compare orbital Rokhlin entropy to Rokhlin entropy, and using an important recent result of Seward, we show that they coincide for free ergodic actions of any countable group. Finally, we consider actions of non-abelian free groups and demonstrate the geometric significance of the entropy equipartition property implied by the Shannon–McMillan–Breiman theorem. We show that the orbital entropy of a partition is the limit of the information functions of the sequence of partitions arising from refining any given finite partition along almost every horoball in the group.

Extending free actions of finite groups on surfaces

The present work introduces new perspectives in order to extend orientation-preserving finite group actions from oriented surfaces to 3-manifolds. We consider the Schur multiplier associated to a finite group G in terms of two-dimensional principal G-bordisms, called G-cobordisms. We are interested in the question of when a free action of a finite group on a closed oriented surface extends to a non-necessarily free action on a 3-manifold. We show the answer to this question is always affirmative for abelian, dihedral, symmetric and alternating groups. As an application of our methods, we show some particular cases for non-necessarily free actions of abelian groups and dihedral groups on surfaces.

Measurable Hall’s theorem for actions of abelian groups

We prove a measurable version of the Hall marriage theorem for actions of finitely generated abelian groups. In particular, it implies that for free measure-preserving actions of such groups and measurable sets which are suitably equidistributed with respect to the action, if they are are equidecomposable, then they are equidecomposable using measurable pieces. The latter generalizes a recent result of Grabowski, Máthé and Pikhurko on the measurable circle squaring and confirms a special case of a conjecture of Gardner.

Congruences on the partial automorphism monoid of a free group action

We study congruences on the partial automorphism monoid of a finite rank free group action. We determine a decomposition of a congruence on this monoid into a Rees congruence, a congruence on a Brandt semigroup and an idempotent separating congruence. The constituent parts are further described in terms of subgroups of direct and semidirect products of groups. We utilize this description to demonstrate how the number of congruences on the partial automorphism monoid depends on the group and the rank of the action.

Dynamic asymptotic dimension for actions of virtually cyclic groups

We show that the dynamic asymptotic dimension of an action of an infinite virtually cyclic group on a compact Hausdorff space is always one if the action has the marker property. This in particular covers a well-known result of Guentner, Willett, and Yu for minimal free actions of infinite cyclic groups. As a direct consequence, we substantially extend a famous result by Toms and Winter on the nuclear dimension of $C^{*}$-algebras arising from minimal free $\mathbb {Z}$-actions. Moreover, we also prove the marker property for all free actions of countable groups on finite-dimensional compact Hausdorff spaces, generalizing a result of Szabó in the metrisable setting.

Free minimal actions of solvable Lie groups which are not affable

We construct an uncountable family of transversely Cantor laminations of compact spaces defined by free minimal actions of solvable groups, which are not affable and whose orbits are not quasi-isometric to Cayley graphs.

Nuclear dimension of simple mathrm {C}^*-algebras

We compute the nuclear dimension of separable, simple, unital, nuclear, {mathcal {Z}}-stable mathrm {C}^*-algebras. This makes classification accessible from {mathcal {Z}}-stability and in particular brings large classes of mathrm {C}^*-algebras associated to free and minimal actions of amenable groups on finite dimensional spaces within the scope of the Elliott classification programme.

Free cyclic group actions on highly-connected 2n-manifolds

Abstract In this paper we study smooth orientation-preserving free actions of the cyclic group ℤ / m {\mathbb{Z}/m} on a class of ( n - 1 ) {(n-1)} -connected 2 ⁢ n {2n} -manifolds, ♯ ⁡ ♯ ⁡ g ⁢ ( S n × S n ) Σ {\mathbin{\sharp}g(S^{n}\times S^{n})\mathbin{\sharp}\Sigma} , where Σ is a homotopy 2 ⁢ n {2n} -sphere. When n = 2 {n=2} , we obtain a classification up to topological conjugation. When n = 3 {n=3} , we obtain a classification up to smooth conjugation. When n ≥ 4 {n\geq 4} , we obtain a classification up to smooth conjugation when the prime factors of m are larger than a constant C ⁢ ( n ) {C(n)} .

The first Pontrjagin classes of homotopy complex projective spaces

Let M2n be an oriented closed smooth manifold homotopy equivalent to the complex projective space CP(n). The main purpose of this paper is to show that when n is even, the difference between the first Pontrjagin class of M2n and that of CP(n) is divisible by 16. As a geometric application of this result, we prove that the Kervaire sphere of dimension 4k+1 does not admit any free circle group action if k+1 is not a power of 2.

Dimension, comparison, and almost finiteness

We develop a dynamical version of some of the theory surrounding the Toms–Winter conjecture for simple separable nuclear C*-algebras and study its connections to the C*-algebra side via the crossed product. We introduce an analogue of hyperfiniteness for free actions of amenable groups on compact spaces and show that it plays the role of \mathcal Z -stability in the Toms–Winter conjecture in its relation to dynamical comparison, and also that it implies \mathcal Z -stability of the crossed product. This property, which we call almost finiteness, generalizes Matui’s notion of the same name from the zero-dimensional setting.We also introduce a notion of tower dimension as a partial analogue of nuclear dimension and study its relation to dynamical comparison and almost finiteness, as well as to the dynamical asymptotic dimension and amenability dimension of Guentner, Willett, and Yu.

Optimized CSIDH Implementation Using a 2-Torsion Point

The implementation of isogeny-based cryptography mainly use Montgomery curves, as they offer fast elliptic curve arithmetic and isogeny computation. However, although Montgomery curves have efficient 3- and 4-isogeny formula, it becomes inefficient when recovering the coefficient of the image curve for large degree isogenies. Because the Commutative Supersingular Isogeny Diffie-Hellman (CSIDH) requires odd-degree isogenies up to at least 587, this inefficiency is the main bottleneck of using a Montgomery curve for CSIDH. In this paper, we present a new optimization method for faster CSIDH protocols entirely on Montgomery curves. To this end, we present a new parameter for CSIDH, in which the three rational two-torsion points exist. By using the proposed parameters, the CSIDH moves around the surface. The curve coefficient of the image curve can be recovered by a two-torsion point. We also proved that the CSIDH while using the proposed parameter guarantees a free and transitive group action. Additionally, we present the implementation result using our method. We demonstrated that our method is 6.4% faster than the original CSIDH. Our works show that quite higher performance of CSIDH is achieved while only using Montgomery curves.

Free abelian group actions on normal projective varieties: submaximal dynamical rank case

AbstractLet X be a normal projective variety of dimension n and G an abelian group of automorphisms such that all elements of $G\setminus \{\operatorname {id}\}$ are of positive entropy. Dinh and Sibony showed that G is actually free abelian of rank $\le n - 1$ . The maximal rank case has been well understood by De-Qi Zhang. We aim to characterize the pair $(X, G)$ such that $\operatorname {rank} G = n - 2$ .

On topological full groups of $\mathbb Z^d$-actions

We give new examples of simple finitely generated groups arising from actions of free abelian groups on the Cantor sets. As particular examples, we discuss groups of interval exchange transformations, and a group naturally associatedwith the Penrose tilings. Many groups in this class are amenable.

Free minimal actions of countable groups with invariant probability measures

We prove that for any countable group $\unicode[STIX]{x1D6E4}$, there exists a free minimal continuous action $\unicode[STIX]{x1D6FC}:\unicode[STIX]{x1D6E4}\curvearrowright {\mathcal{C}}$ on the Cantor set admitting an invariant Borel probability measure.

A classification of certain codimension one locally free actions of nilpotent Lie groups up to a differentiable orbital conjugacy

Let G be a simply connected nilpotent Lie group, let M be a compact connected manifold with dim(M) = dim(G)+1 and let Φ be a C∞ locally free action of G on M. If G admits no lattice and if the centralizer in G of its derived group G′ is the center of G′, then Φ is differentiably orbitally conjugated to a homogeneous action of the same kind.

Ping-pong configurations and circular orders on free groups

We discuss actions of free groups on the circle with “ping-pong” dynamics; these are dynamics determined by a finite amount of combinatorial data, analogous to Schottky domains orMarkov partitions. Using this, we show that the free group F_n admits an isolated circular order if and only if n is even, in stark contrast with the case for linear orders. This answers a question from [21]. Inspired by work in [2], we also exhibit examples of “exotic” isolated points in the space of all circular orders on F_2 . Analogous results are obtained for linear orders on the groups F_n \times \mathbb Z .

Almost Finiteness and the Small Boundary Property

Working within the framework of free actions of countable amenable groups on compact metrizable spaces, we show that the small boundary property is equivalent to a density version of almost finiteness, which we call almost finiteness in measure, and that under this hypothesis the properties of almost finiteness, comparison, and $m$-comparison for some nonnegative integer $m$ are all equivalent. The proof combines an Ornstein-Weiss tiling argument with the use of zero-dimensional extensions which are measure-isomorphic over singleton fibres. These kinds of extensions are also employed to show that if every free action of a given group on a zero-dimensional space is almost finite then so are all free actions of the group on spaces with finite covering dimension. Combined with recent results of Downarowicz-Zhang and Conley-Jackson-Marks-Seward-Tucker-Drob on dynamical tilings and of Castillejos-Evington-Tikuisis-White-Winter on the Toms-Winter conjecture, this implies that crossed product C$^*$-algebras arising from free minimal actions of groups with local subexponential growth on finite-dimensional spaces are classifiable in the sense of Elliott's program. We show furthermore that, for free actions of countably infinite amenable groups, the small boundary property implies that the crossed product has uniform property $\Gamma$, which under minimality confirms the Toms-Winter conjecture for such crossed products by the aforementioned work of Castillejos-Evington-Tikuisis-White-Winter.

C⁎-simplicity and representations of topological full groups of groupoids

Given an ample groupoid G with compact unit space, we study the canonical representation of the topological full group [[G]] in the full groupoid C⁎-algebra C⁎(G). In particular, we show that the image of this representation generates C⁎(G) if and only if C⁎(G) admits no tracial state. The techniques that we use include the notion of groups covering groupoids.As an application, we provide sufficient conditions for C⁎-simplicity of certain topological full groups, including those associated with topologically free and minimal actions of non-amenable and countable groups on the Cantor set.

Uniform exponential growth for CAT(0) square complexes

In this paper we start the inquiry into proving uniform exponential growth in the context of groups acting on CAT(0) cube complexes. We address free group actions on CAT(0) square complexes and prove a more general statement. This says that if $F$ is a finite collection of hyperbolic automorphisms of a CAT(0) square complex $X$, then either there exists a pair of words of length at most 10 in $F$ which freely generate a free semigroup, or all elements of $F$ stabilize a flat (of dimension 1 or 2 in $X$). As a corollary, we obtain a lower bound for the growth constant, $\sqrt[10]{2}$, which is uniform not just for a given group acting freely on a given CAT(0) cube complex, but for all groups which are not virtually abelian and have a free action on a CAT(0) square complex.

$\mathbb{Z}^{d}$-odometers and cohomology

Cohomology for actions of free abelian groups on the Cantor set has (when endowed with an order structure) provided a complete invariant for orbit equivalence. In this paper, we study a particular class of actions of such groups called odometers (or profinite actions) and investigate their cohomology. We show that for a free, minimal \mathbb Z^d -odometer, the first cohomology group provides a complete invariant for the action up to conjugacy. This is in contrast with the situation for orbit equivalence where it is the cohomology in dimension d which provides the invariant. We also consider classification up to isomorphism and continuous orbit equivalence.