G-equivariant Map Research Articles

Group Actions on Twisted Sums of Banach Spaces

We study bounded actions of groups and semigroups G on exact sequences of Banach spaces from the point of view of (generalized) quasilinear maps, characterize the actions on the twisted sum space by commutator estimates and introduce the associated notions of G-centralizer and G-equivariant map. We will show that when (A) G is an amenable group and (U) the target space is complemented in its bidual by a G-equivariant projection, then uniformly bounded compatible families of operators generate bounded actions on the twisted sum space; that compatible quasilinear maps are linear perturbations of G-centralizers; and that, under (A) and (U), G-centralizers are bounded perturbations of G-equivariant maps. The previous results are optimal. Several examples and counterexamples are presented involving the action of the isometry group of L_p(0,1), pne 2 on the Kalton–Peck space Z_p, certain non-unitarizable triangular representations of the free group {mathbb {F}}_infty on the Hilbert space, the compatibility of complex structures on twisted sums, or bounded actions on the interpolation scale of L_p-spaces. In the penultimate section we consider the category of G-Banach spaces and study its exact sequences, showing that, under (A) and (U), G-splitting and usual splitting coincide. The purpose of the final section is to present some applications, showing that several previous result are optimal and to suggest further open lines of research.

The Bestvina–Edwards theorem and the Hilbert–Smith conjecture

We prove a number of results surrounding the Borsuk–Ulam-type conjecture of Baum, Dabrowski, and Hajac (BDH, for short), which states that given a free action of a compact group G on a compact space X, there are no G-equivariant maps X∗G→X (with ∗ denoting the topological join). Mainly, we prove the BDH conjecture for locally trivial principal G-bundles. The proof relies on the nonexistence of G-equivariant maps G∗(n+1)→G∗n, which in turn is a strengthening of an unpublished result of Bestvina and Edwards. Moreover, we show that the BDH conjecture partially settles a conjecture of Ageev which implies the weak version of the Hilbert–Smith conjecture stating that no infinite compact zero-dimensional group can act freely on a manifold so that the orbit space is finite-dimensional.

Gromov’s Oka Principle for Equivariant Maps

We take the first step in the development of an equivariant version of modern, Gromov-style Oka theory. We define equivariant versions of the standard Oka property, ellipticity, and homotopy Runge property of complex manifolds, show that they satisfy all the expected basic properties, and present examples. Our main theorem is an equivariant Oka principle saying that if a finite group G acts on a Stein manifold X and another manifold Y in such a way that Y is G-Oka, then every G-equivariant continuous map \(X\rightarrow Y\) can be deformed, through such maps, to a G-equivariant holomorphic map. Approximation on a G-invariant holomorphically convex compact subset of X and jet interpolation along a G-invariant subvariety of X can be built into the theorem. We conjecture that the theorem holds for actions of arbitrary reductive complex Lie groups and prove partial results to this effect.

The category of G-posets

For a group G and a poset X, we say that X is a G-poset if it is equipped with a G-action that is monotone. If we consider the mononote G-equivariant maps as morphisms, then we get the category of G-posets. We give a description of the projective objects, injective objects, quotients, and generators.

Dold’s theorem from viewpoint of strong compatibility graphs

Let G be a non-trivial finite group. The well-known Dold’s theorem states that: There is no continuous G-equivariant map from an n-connected simplicial G-complex to a free simplicial G-complex of dimension at most n. In this paper, we give a new generalization of Dold’s theorem, by replacing “dimension at most n” with a sharper combinatorial parameter. Indeed, this parameter is the chromatic number of a new family of graphs, called strong compatibility graphs, associated to the target space. Moreover, in a series of examples, we will see that one can hope to infer much more information from this generalization than ordinary Dold’s theorem. In particular, we show that this new parameter is significantly better than the dimension of target space “for almost all free Z2-simplicial complex.” Finally, as another application of strong compatibility graphs, some new results on the limitations of topological methods for determining the chromatic number of graphs will be presented.

Crossed products of operator systems

In this paper we introduce the crossed product construction for a discrete group action on an operator system. In analogy to the work of E. Katsoulis and C. Ramsey, we describe three canonical crossed products arising from such a dynamical system. We describe how these crossed product constructions behave under G-equivariant maps, tensor products, and the canonical C⁎-covers. We show that hyperrigidity is preserved under two of the three crossed products. Finally, using A. Kavruk's notion of an operator system that detects C⁎-nuclearity, we give a negative answer to a question on operator algebra crossed products posed by Katsoulis and Ramsey.

General Bourgin–Yang theorems

We describe a unified approach to estimating the dimension of f−1(A) for any G-equivariant map f:X→Y and any closed G-invariant subset A⊆Y in terms of connectivity of X and dimension of Y, where G is either a cyclic group of order pk, a p-torus (p a prime), or a torus.

Parametrized Borsuk–Ulam theorems for free involutions on [formula omitted

Let X be a finitistic space with mod 2 cohomology algebra isomorphic to that of FPm×S3, where F=R,C or H. Let (X,E,π,B) be a fibre bundle and (Rk,E′,π′,B) be a k-dimensional real vector bundle with fibre preserving G=Z2 action such that G acts freely on E and E′−{0}, where {0} is the zero section of the vector bundle. We determine lower bounds for the cohomological dimension of the zero set f−1({0}) of a fibre preserving G-equivariant map f:E→E′. As an application of this result, we determine a lower bound for the cohomological dimension of the coincidence sets of continuous maps f:X→Rn. In particular, we estimate the size of the coincidence sets of continuous maps f:Si×S3→Rk relative to any free involution on Si×S3, (i=1,2,4).

Semi-direct products of Lie algebras and covariants

Let Q be a connected algebraic group with Lie algebra q. Symmetric invariants of q, i.e., the Q-invariants in the symmetric algebra S(q) of q, is a first approximation to the understanding of the coadjoint action (Q:q⁎) and coadjoint Q-orbits. In this article, we study a class of non-reductive Lie algebras, where the description of the symmetric invariants is possible and the coadjoint representation has a number of nice invariant-theoretic properties. If G is a semisimple group with Lie algebra g and V is G-module, then we define q to be the semi-direct product of g and V. Then we are interested in the case, where the generic isotropy group for the G-action on V is reductive and commutative. It turns out that in this case symmetric invariants of q can be constructed via certain G-equivariant maps from g to V (“covariants”).

Borsuk-Ulam Theorems and Their Parametrized Versions for $$\mathbb {F}P^m\times \mathbb {S}^3$$ F P m × S 3

Let $$G=\mathbb {Z}_p,$$ $$p>2$$ a prime, act freely on a finitistic space X with mod p cohomology ring isomorphic to that of $$\mathbb {F}P^m\times \mathbb {S}^3$$ , where $$m+1\not \equiv 0$$ mod p and $$\mathbb {F}=\mathbb {C}$$ or $$\mathbb {H}$$ . We wish to discuss the nonexistence of G-equivariant maps $$\mathbb {S}^{2q-1}\rightarrow X$$ and $$ X\rightarrow \mathbb {S}^{2q-1}$$ , where $$\mathbb {S}^{2q-1}$$ is equipped with a free G-action. These results are analogues of the celebrated Borsuk-Ulam theorem. To establish these results first we find the cohomology algebra of orbit spaces of free G-actions on X. For a continuous map $$f\!:\! X\rightarrow \mathbb {R}^n$$ , a lower bound of the cohomological dimension of the partial coincidence set of f is determined. Furthermore, we approximate the size of the zero set of a fibre preserving G-equivariant map between a fibre bundle with fibre X and a vector bundle. An estimate of the size of the G-coincidence set of a fibre preserving map is also obtained. These results are parametrized versions of the Borsuk-Ulam theorem.

I. GM-sets Applied to Lie theory

This paper is devoted to GM-sets and their applications in Lie theory from the theoretical and computational points of view. We consider the GM-set of a transformation group G acting on a smooth manifold M as the set of all associated G-equivariant maps and then investigate the relation between the concepts of G-conjugacy and G-invariance in Lie theory with respect to the associated GM-set. We will also have a look at the fundamental and constructive concept of moving frames through the GM-sets and apply this construction to carry out a detailed survey of some basic concepts in Lie algebra theory.

Links between generalized Montréal-functors

Let $o$ be the ring of integers in a finite extension $K/\mathbb{Q}_p$ and $G=\mathbf{G}(\mathbb{Q}_p)$ be the $\mathbb{Q}_p$-points of a $\mathbb{Q}_p$-split reductive group $\mathbf{G}$ defined over $\mathbb{Z}_p$ with connected centre and split Borel $\mathbf{B}=\mathbf{TN}$. We show that Breuil's pseudocompact $(\varphi,\Gamma)$-module $D^\vee_{\xi}(\pi)$ attached to a smooth $o$-torsion representation $\pi$ of $B=\mathbf{B}(\mathbb{Q}_p)$ is isomorphic to the pseudocompact completion of the basechange $\mathcal{O_E}\otimes_{\Lambda(N_0),\ell}\widetilde{D_{SV}}(\pi)$ to Fontaine's ring (via a Whittaker functional $\ell\colon N_0=\mathbf{N}(\mathbb{Z}_p)\to \mathbb{Z}_p$) of the \'etale hull $\widetilde{D_{SV}}(\pi)$ of $D_{SV}(\pi)$ defined by Schneider and Vigneras. Moreover, we construct a $G$-equivariant map from the Pontryagin dual $\pi^\vee$ to the global sections $\mathfrak{Y}(G/B)$ of the $G$-equivariant sheaf $\mathfrak{Y}$ on $G/B$ attached to a noncommutative multivariable version $D^\vee_{\xi,\ell,\infty}(\pi)$ of Breuil's $D^\vee_{\xi}(\pi)$ whenever $\pi$ comes as the restriction to $B$ of a smooth, admissible representation of $G$ of finite length.

Equivariant unirationality of del Pezzo surfaces of degree 3 and 4

A variety X with an action of a finite group G is said to be G-unirational if there is a G-equivariant dominant rational map $$V \dasharrow X$$ where V is a faithful linear representation of G. This generalizes the usual notion of unirationality. We determine when X is G-unirational for any complex del Pezzo surface X of degree at least 3.

Multivariable $$(\varphi ,\Gamma )$$ ( φ , Γ ) -modules and smooth o-torsion representations

Let G be a $$\mathbb {Q}_p$$ -split reductive group with connected centre and Borel subgroup $$B=TN$$ . We construct a right exact functor $$D^\vee _\Delta $$ from the category of smooth modulo $$p^n$$ representations of B to the category of projective limits of finitely generated étale $$(\varphi ,\Gamma )$$ -modules over a multivariable (indexed by the set of simple roots) commutative Laurent series ring. These correspond to representations of a direct power of $$\mathrm {Gal}(\overline{\mathbb {Q}_p}/\mathbb {Q}_p)$$ via an equivalence of categories. Parabolic induction from a subgroup $$P=L_PN_P$$ gives rise to a basechange from a Laurent series ring in those variables with corresponding simple roots contained in the Levi component $$L_P$$ . $$D^\vee _\Delta $$ is exact and yields finitely generated objects on the category $$SP_A$$ of finite length representations with subquotients of principal series as Jordan–Hölder factors. Lifting the functor $$D^\vee _\Delta $$ to all (noncommuting) variables indexed by the positive roots allows us to construct a G-equivariant sheaf $$\mathfrak {Y}_{\pi ,\Delta }$$ on G / B and a G-equivariant continuous map from the Pontryagin dual $$\pi ^\vee $$ of a smooth representation $$\pi $$ of G to the global sections $$\mathfrak {Y}_{\pi ,\Delta }(G/B)$$ . We deduce that $$D^\vee _\Delta $$ is fully faithful on the full subcategory of $$SP_A$$ with Jordan–Hölder factors isomorphic to irreducible principal series.

Examples of algebraically realized maps

Let \(G= {\mathbb {Z}}_4\). We construct examples of G-equivariant entire rational maps from non-singular real algebraic G-varieties to Grassmannians with appropriate actions of G. These examples of strongly algebraic \({\mathbb {Z}}_4\) vector bundles facilitate a key step in the verification of Conjecture 1.1 in the general cyclic group action case.

Equivariant Maps from Stiefel Bundles to Vector Bundles

AbstractLet E → B be a fibre bundle and let Eʹ → B be a vector bundle. Let G be a compact Lie group acting fibre preservingly and freely on both E and Eʹ – 0, where 0 is the zero section of Eʹ → B. Let f : E → Eʹ be a fibre-preserving G-equivariant map and let Zf = {x ∈ E | f(x) = 0} be the zero set of f. In this paper we give a lower bound for the cohomological dimension of the zero set Zf when a fibre of E → B is a real Stiefel manifold with a free ℤ/2-action or a complex Stiefel manifold with a free 𝕊1-action. This generalizes a well-known result of Dold for sphere bundles equipped with free involutions.

Zero sets of equivariant maps from products of spheres to Euclidean spaces

Let E→B be a fiber bundle and E′→B be a vector bundle. Let G be a compact group acting fiber preservingly and freely on both E and E′−0, where 0 is the zero section of E′→B. Let f:E→E′ be a fiber preserving G-equivariant map, and let Zf={x∈E | f(x)=0} be the zero set of f. It is an interesting problem to estimate the dimension of the set Zf. In 1988, Dold [5] obtained a lower bound for the cohomological dimension of the zero set Zf when E→B is the sphere bundle associated with a vector bundle which is equipped with the antipodal action of G=Z/2. In this paper, we generalize this result to products of finitely many spheres equipped with the diagonal antipodal action of Z/2. We also prove a Bourgin–Yang type theorem for products of spheres equipped with the diagonal antipodal action of Z/2.

Borel structurability on the 2-shift of a countable group

We show that for any infinite countable group G and for any free Borel action G↷X there exists an equivariant class-bijective Borel map from X to the free part Free(2G) of the 2-shift G↷2G. This implies that any Borel structurability which holds for the equivalence relation generated by G↷Free(2G) must hold a fortiori for all equivalence relations coming from free Borel actions of G. A related consequence is that the Borel chromatic number of Free(2G) is the maximum among Borel chromatic numbers of free actions of G. This answers a question of Marks. Our construction is flexible and, using an appropriate notion of genericity, we are able to show that in fact the generic G-equivariant map to 2G lands in the free part. As a corollary we obtain that for every ϵ>0, every free p.m.p. action of G has a free factor which admits a 2-piece generating partition with Shannon entropy less than ϵ. This generalizes a result of Danilenko and Park.

Boundaries of reduced C*C^{*}-algebras of discrete groups

Abstract For a discrete group G, we consider the minimal C * C^{*} -subalgebra of ℓ ∞ \ell^{\infty} (G) that arises as the image of a unital positive G-equivariant projection. This algebra always exists and is unique up to isomorphism. It is trivial if and only if G is amenable. We prove that, more generally, it can be identified with the algebra C( ∂ F ⁡ G \partial_{F}G ) of continuous functions on Furstenberg’s universal G-boundary ∂ F ⁡ G {\partial_{F}G} . This operator-algebraic construction of the Furstenberg boundary has a number of interesting consequences. We prove that G is exact precisely when the G-action on ∂ F ⁡ G {\partial_{F}G} is amenable, and use this fact to prove Ozawa’s conjecture that if G is exact, then there is an embedding of the reduced C * C^{*} -algebra C r * ⁢ ( G ) {\mathrm{C}^{*}_{r}(G)} of G into a nuclear C * C^{*} -algebra which is contained in the injective envelope of C r * ⁢ ( G ) {\mathrm{C}^{*}_{r}(G)} . The algebra C( ∂ F ⁡ G \partial_{F}G ) arises as an injective envelope in the sense of Hamana, which implies rigidity results for certain G-equivariant maps. We prove a generalization of a rigidity result of Ozawa for G-equivariant maps between spaces of functions on the hyperbolic boundary of a hyperbolic group. Our result applies to hyperbolic groups, but also to groups that are not hyperbolic or even relatively hyperbolic, including certain mapping class groups. It is a longstanding open problem to determine which groups are C * C^{*} -simple, in the sense that the algebra C r * ⁢ ( G ) {\mathrm{C}^{*}_{r}(G)} is simple. We prove that this problem can be reformulated as a problem about the structure of the G-action on the Furstenberg boundary. Specifically, we prove that a discrete group G is C * C^{*} -simple if and only if the G-action on the Furstenberg boundary is topologically free. We apply this result to prove that Tarski monster groups are C * C^{*} -simple. This provides another solution to a problem of de la Harpe (recently answered by Olshanskii and Osin) about the existence of C * C^{*} -simple groups with no free subgroups.

On the existence of G-equivariant maps

Let G be a compact Lie group. Let X, Y be free G-spaces. In this paper, by using the numerical index i (X; R), under cohomological conditions on the spaces X and Y, we consider the question of the existence of G-equivariant maps f: X → Y.