Non-free Actions Research Articles

Nonfree almost finite actions for locally finite‐by‐virtually Z${\mathbb {Z}}$ groups

AbstractIn this paper, we study almost finiteness and almost finiteness in measure of nonfree actions. Let be a minimal action of a locally finite‐by‐virtually group on the Cantor set . We prove that under certain assumptions, the action is almost finite in measure if and only if is essentially free. As an application, we obtain that any minimal topologically free action of a virtually group on an infinite compact metrizable space with the small boundary property is almost finite. This is the first general result, assuming only topological freeness, in this direction, and these lead to new results on uniform property and ‐stability for their crossed product ‐algebras. Some concrete examples of minimal topological free (but nonfree) subshifts are provided.

Examples of surfaces with canonical maps of degree 12, 13, 15, 16 and 18

In this note, we present examples of complex algebraic surfaces with canonical maps of degree 12, 13, 15, 16 and 18. They are constructed as quotients of a product of two curves of genus 10 and 19 using certain non-free actions of the group S3×Z32\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S_3\ imes {\\mathbb {Z}}_3^2$$\\end{document}. To our knowledge, there are no other examples in the literature of surfaces with canonical map of degree 13, 15 and 18.

On Nonfree Actions of Commuting Involutions on Manifolds

On Nonfree Actions of Commuting Involutions on Manifolds

On Nonfree Actions of Commuting Involutions on Manifolds

Получена новая нижняя оценка, связывающая рациональную когомологическую длину базы и тотального пространства разветвленных накрытий ориентируемых многообразий в случае, когда разветвленное накрытие является проекцией на факторпространство по действию коммутирующих инволюций на тотальном пространстве. Эта оценка существенно сильнее классической оценки Берстейна-Эдмондса 1978 года, справедливой для произвольных разветвленных накрытий ориентируемых многообразий. В рамках теории разветвленных накрытий получены результаты, мотивированные проблематикой $n$-значных топологических групп. Мы в явном виде строим $m-1$ коммутирующих инволюций, действующих автоморфизмами на торе $T^m$, с пространством орбит $\mathbb{R}P^m$ для любого нечетного $m\ge 3$. В силу полученной конструкции многообразие $\mathbb{R}P^m$ несет структуру $2^{m-1}$-значной абелевой топологической группы для всех нечетных $m\ge 3$. Библиография: 7 названий.

Almost finiteness and homology of certain non-free actions

We show that Cantor minimal \mathbb{Z}\rtimes\mathbb{Z}_2 -systems and essentially free amenable odometers are almost finite. We also compute the homology groups of Cantor minimal \mathbb{Z}\rtimes\mathbb{Z}_2 -systems and show that the associated transformation groupoids satisfy the HK conjecture if and only if the action is free.

Stratification of singular hyperkähler quotients

Abstract Hyperkähler quotients by non-free actions are typically singular, but are nevertheless partitioned into smooth hyperkähler manifolds. We show that these partitions are topological stratifications, in a strong sense. We also endow the quotients with global Poisson structures which recover the hyperkähler structures on the strata. Finally, we give a local model which shows that these quotients are locally isomorphic to linear complex-symplectic reductions in the GIT sense. These results can be thought of as the hyperkähler analogues of Sjamaar–Lerman’s theorems for singular symplectic reduction. They are based on a local normal form for the underlying complex-Hamiltonian manifold, which may be of independent interest.

Optimal colored Tverberg theorems for prime powers

The type A colored Tverberg theorem of Blagojevic, Matschke, and Ziegler provides optimal bounds for the colored Tverberg problem, under the condition that the number of intersecting rainbow simplices is a prime number. We extend this result to an optimal, type A colored Tverberg theorem for multisets of colored points, which is valid for each prime power $r=p^k$. One of the principal new ideas is to replace the ambient $\Delta^N$, used in the original Tverberg theorem, by an abridged simplex of smaller dimension, and to compensate for this reduction by allowing vertices to repeatedly appear a controlled number of times in different rainbow simplices. Configuration spaces, used in the proof, are combinatorial pseudomanifolds which can be represented as multiple chessboard complexes. Our main topological tool is the Eilenberg-Krasnoselskii theory of degrees of equivariant maps for non-free actions.

The nuclear dimension of C⁎-algebras associated to topological flows and orientable line foliations

We show that for any locally compact Hausdorff space Y with finite covering dimension and for any continuous flow R↷Y, the resulting crossed product C⁎-algebra C0(Y)⋊R has finite nuclear dimension. This generalizes previous results for free flows, where this was proved using Rokhlin dimension techniques. As an application, we obtain bounds for the nuclear dimension of C⁎-algebras associated to one-dimensional orientable foliations. This result is analogous to the one we obtained earlier for non-free actions of Z. Some novel techniques in our proof include the use of a conditional expectation constructed from the inclusion of a clopen subgroupoid, as well as the introduction of what we call fiberwise groupoid coverings that help us build a link between foliation C⁎-algebras and crossed products.

On diagonal actions of branch groups and the corresponding characters

We introduce notions of absolutely non-free and perfectly non-free group actions and use them to study the associated unitary representations. We show that every weakly branch group acting on a regular rooted tree acts absolutely non-freely on the boundary of the tree. Using this result and the symmetrized diagonal actions we construct for every countable branch group infinitely many different ergodic perfectly non-free actions, infinitely many II1-factor representations, and infinitely many continuous ergodic invariant random subgroups.

On purely non-free finite actions of abelian groups on compact surfaces

A finite group of self-homeomorphisms of a closed orientable surface is said to act on it purely non-freely if each of its elements has a fixed point; we also call it a gpnf-action. In this paper we observe that gpnf-actions exist for an arbitrary finite group and we discuss the minimum genus problem for such actions. We solve it for abelian groups. In the cyclic case we prove that the minimal gpnf-action genus coincides with Harvey’s minimal genus.

On equivariant and invariant topological complexity of smooth ℤ/_{𝕡}-spheres

We investigate equivariant and invariant topological complexity of spheres endowed with smooth non-free actions of cyclic groups of prime order. We prove that semilinear Z / p \mathbb {Z}/_{\!p} -spheres have both invariants either 2 2 or 3 3 and calculate exact values in all but two cases. On the other hand, we exhibit examples which show that these invariants can be arbitrarily large in the class of smooth Z / p \mathbb {Z}/_{\!p} -spheres.

SOME EXOTIC ACTIONS OF FINITE GROUPS ON SMOOTH 4-MANIFOLDS

Using $G$-monopole invariants, we produce infinitely many exotic non-free actions of $\mathbb{Z}_{k}\oplus H$ on some connected sums of finite number of $S^{2}\times S^{2}$, $\mathbb{C}P_{2}$, $\overline{\mathbb{C}P}_{2}$, and $K3$ surfaces, where $k\geq 2$, and $H$ is any nontrivial finite group acting freely on $S^{3}$.

Weak equivalence and non-classifiability of measure preserving actions

AbstractAbért and Weiss have shown that the Bernoulli shift ${s}_{\Gamma } $ of a countably infinite group $\Gamma $ is weakly contained in any free measure preserving action $\boldsymbol{a}$ of $\Gamma $. Proving a conjecture of Ioana, we establish a strong version of this result by showing that ${\boldsymbol{s}}_{\Gamma } \times \boldsymbol{a}$ is weakly equivalent to $\boldsymbol{a}$. Using random Bernoulli shifts introduced by Abért, Glasner, and Virag, we generalize this to non-free actions, replacing ${\boldsymbol{s}}_{\Gamma } $ with a random Bernoulli shift associated to an invariant random subgroup, and replacing the product action with a relatively independent joining. The result for free actions is used along with the theory of Borel reducibility and Hjorth’s theory of turbulence to show that, on the weak equivalence class of a free measure preserving action, the equivalence relations of isomorphism, weak isomorphism, and unitary equivalence are not classifiable by countable structures. This in particular shows that there are no free weakly rigid actions, that is, actions whose weak equivalence class and isomorphism class coincide, answering negatively a question of Abért and Elek. We also answer a question of Kechris regarding two ergodic theoretic properties of residually finite groups. A countably infinite residually finite group $\Gamma $ is said to have property ${\text{EMD} }^{\ast } $ if the action ${\boldsymbol{p}}_{\Gamma } $ of $\Gamma $ on its profinite completion weakly contains all ergodic measure preserving actions of $\Gamma $, and $\Gamma $ is said to have property $\text{MD} $ if $\boldsymbol{\iota} \times {\boldsymbol{p}}_{\Gamma } $ weakly contains all measure preserving actions of $\Gamma $, where $\boldsymbol{\iota} $ denotes the identity action on a standard non-atomic probability space. Kechris has shown that ${\text{EMD} }^{\ast } $ implies $\text{MD} $ and asked if the two properties are actually equivalent. We provide a positive answer to this question by studying the relationship between convexity and weak containment in the space of measure preserving actions.

$$G$$ -monopole classes, Ricci flow, and Yamabe invariants of 4-manifolds

On a smooth closed oriented 4-manifold \(M\) with a smooth action by a finite group \(G\), we show that a \(G\)-monopole class gives the \(L^2\)-estimate of the Ricci curvature of a \(G\)-invariant Riemannian metric, and derive a topological obstruction to the existence of a \(G\)-invariant nonsingular solution to the normalized Ricci flow on \(M\). In particular, for certain \(m\) and \(n, m\mathbb C P_2 \# n\overline{\mathbb{C P}}_2\) admits an infinite family of topologically equivalent but smoothly distinct non-free actions of \(\mathbb Z _d\) such that it admits no nonsingular solution to the normalized Ricci flow for any initial metric invariant under such an action, where \(d>1\) is a non-prime integer. We also compute the \(G\)-Yamabe invariants of some 4-manifolds with \(G\)-monopole classes and the orbifold Yamabe invariants of some 4-orbifolds.

Totally Nonfree Actions and the Infinite Symmetric Group

We consider the totally nonfree (TNF) action of a groups and the corresponding adjoint invariant (AD) measures on the lattices of the subgroups of the given group. The main result is the description of all adjoint-invariant and TNF-measures on the lattice of subgroups of the infinite symmetric group SN. The problem is closely related to the theory of characters and factor representations of groups.

Homotopy classification of actions of finite groups on aspherical spaces

In this paper, actions of finite groups on aspherical spaces are studied. A complete homotopy classification of free actions of finite groups on aspherical spaces is obtained, including some results in nonfree actions. For example, a relation between cohomologies of finite groups and the lattice structure of their subgroups is obtained by this method; this relation is formulated in terms of spectral sequences.

Nonfree actions of countable groups and their characters

We introduce the definition of totally nonfree actions of groups; such actions are naturally related to the theory of characters of groups and their factor-representations of type II. This short note is a brief exposition of a part of a more detailed paper on this subject, which is now in preparation.

Singular reduction of Dirac structures

The regular reduction of a Dirac manifold acted upon freely and properly by a Lie group is generalized to a nonfree action. For this, several facts about G G -invariant vector fields and one-forms are shown.

Proper actions which are not saturated

If a locally compact group G acts properly on a locally compact space X, then the induced action on C 0 (X) is proper in the sense of Rieffel, with generalised fixed-point algebra C 0 (G\X). Rieffel's theory then gives a Morita equivalence between C 0 (G\X) and an ideal I in the crossed product C 0 (X) x G; we identify I by describing the primitive ideals which contain it, and we deduce that I = C 0 (X) x G if and only if G acts freely. We show that if a discrete group G acts on a directed graph E and every vertex of E has a finite stabiliser, then the induced action α of G on the graph C * -algebra C * (E) is proper. When G acts freely on E, the generalised fixed-point algebra C * (E) α is isomorphic to C * (G\E) and Morita equivalent to C * (E) x G, in parallel with the situation for free and proper actions on spaces, but this parallel does not seem to give useful predictions for nonfree actions.

The sectional curvature remains positive when taking quotients by certain nonfree actions

We study some cases when the sectional curvature remains positive under the taking of quotients by certain nonfree isometric actions of Lie groups. We consider the actions of the groups $S^1$ and $S^3$ such that the quotient space can be endowed with a smooth structure using the fibrations $S^3/S^1{\simeq}S^2$ and $S^7/S^3\simeq S^4$. We prove that the quotient space carries a metric of positive sectional curvature, provided that the original metric has positive sectional curvature on all 2-planes orthogonal to the orbits of the action.