Infinite Discrete Group Research Articles

Bauer simplices and the small boundary property

We show that, for every minimal action of a countably infinite discrete group on a compact metrizable space, if the extreme boundary of the simplex of invariant Borel probability measures is closed and has finite covering dimension then the action has the small boundary property.

Variational principle of higher dimension weighted pressure for amenable group actions

Let r≥2 and (Xi,G)(i=1,⋯,r) be topological dynamical systems with an infinite countable discrete amenable phase group G. Suppose that πi:(Xi,G)→(Xi+1,G) are factor maps, a={a1,⋯,ar−1}∈Rr−1 is a vector with 0≤ai≤1 and (w1,⋯,wr)∈Rr is a probability vector associated with a. In this paper, given f∈C(X1), we introduce the weighted topological pressure Pa(f,G). Moreover, by using measure-theoretical theory, we establish a variational principle:Pa(f,G)=supμ∈MG(X1)⁡(∑i=1rwihμi(Xi,G)+w1∫X1fdμ), where h{⋅}(⋅,G) is the Kolmogorov-Sinai entropy of the systems acted by the amenable group G and μi=πi−1∘⋯∘π1μ is the induced G-invariant measure on Xi.

AdS3 orbifolds, BTZ black holes, and holography

Conical defects of the form (AdS3 × S3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathbbm{S}}^3 $$\\end{document})/ℤk have an exact orbifold description in worldsheet string theory, which we derive from their known presentation as gauged Wess-Zumino-Witten models. The configuration of strings and fivebranes sourcing this geometry is well-understood, as is the correspondence to states/operators in the dual CFT2. One can analytically continue the construction to Euclidean AdS3 (i.e. the hyperbolic ball ℍ3+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathbb{H}}_3^{+} $$\\end{document}) and consider the orbifold by any infinite discrete (Kleinian) group generated by a set of elliptic elements γi ∈ SL(2, ℂ), γiki\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\gamma}_i^{{\ extrm{k}}_i} $$\\end{document} = \U0001d7d9, i = 1, . , K. The resulting geometry consists of multiple conical defects traveling along geodesics in ℍ3+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathbb{H}}_3^{+} $$\\end{document}, and provides a semiclassical bulk description of correlation functions in the dual CFT involving the corresponding defect operators, which is nonperturbatively exact in α′. The Lorentzian continuation of these geometries describes a collection of defects colliding to make a BTZ black hole. We comment on a recent proposal to use such correlators to prepare a basis of black hole microstates, and elaborate on a picture of black hole formation and evaporation in terms of the underlying brane dynamics in the bulk.

Proper Equivariant Stable Homotopy Theory

This monograph introduces a framework for genuine proper equivariant stable homotopy theory for Lie groups. The adjective ‘proper’ alludes to the feature that equivalences are tested on compact subgroups, and that the objects are built from equivariant cells with compact isotropy groups; the adjective ‘genuine’ indicates that the theory comes with appropriate transfers and Wirthmüller isomorphisms, and the resulting equivariant cohomology theories support the analog of an R O ( G ) R O(G) -grading. Our model for genuine proper G G -equivariant stable homotopy theory is the category of orthogonal G G -spectra; the equivalences are those morphisms that induce isomorphisms of equivariant stable homotopy groups for all compact subgroups of G G . This class of π ∗ \pi _* -isomorphisms is part of a symmetric monoidal stable model structure, and the associated tensor triangulated homotopy category is compactly generated. Consequently, every orthogonal G G -spectrum represents an equivariant cohomology theory on the category of G G -spaces. These represented cohomology theories are designed to only depend on the ‘proper G G -homotopy type’, tested by fixed points under all compact subgroups. An important special case of our theory are infinite discrete groups. For these, our genuine equivariant theory is related to finiteness properties in the sense of geometric group theory; for example, the G G -sphere spectrum is a compact object in our triangulated equivariant homotopy category if the universal space for proper G G -actions has a finite G G -CW-model. For discrete groups, the represented equivariant cohomology theories on finite proper G G -CW-complexes admit a more explicit description in terms of parameterized equivariant homotopy theory, suitably stabilized by G G -vector bundles. Via this description, we can identify the previously defined G G -cohomology theories of equivariant stable cohomotopy and equivariant K-theory as cohomology theories represented by specific orthogonal G G -spectra.

Relative uniformly positive entropy of induced amenable group actions

AbstractLet G be a countably infinite discrete amenable group. It should be noted that a G-system $(X,G)$ naturally induces a G-system $(\mathcal {M}(X),G)$ , where $\mathcal {M}(X)$ denotes the space of Borel probability measures on the compact metric space X endowed with the weak*-topology. A factor map $\pi : (X,G)\to (Y,G)$ between two G-systems induces a factor map $\widetilde {\pi }:(\mathcal {M}(X),G)\to (\mathcal {M}(Y),G)$ . It turns out that $\widetilde {\pi }$ is open if and only if $\pi $ is open. When Y is fully supported, it is shown that $\pi $ has relative uniformly positive entropy if and only if $\widetilde {\pi }$ has relative uniformly positive entropy.

Dynamics of actions of automorphisms of discrete groups $G$ on $\mathrm{Sub}_G$ and applications to lattices in Lie groups

For a locally compact Hausdorff group G and the compact space \mathrm{Sub}_G of closed subgroups of G endowed with the Chabauty topology, we study the dynamics of actions of automorphisms of G on \mathrm{Sub}_G in terms of distality and expansivity. We prove that an infinite discrete group G , which is either polycyclic or a lattice in a connected Lie group, does not admit any automorphism which acts expansively on \mathrm{Sub}^c_G , the space of cyclic subgroups of G , while only the finite order automorphisms of G act distally on \mathrm{Sub}^c_G . For an automorphism T of a connected Lie group G which keeps a lattice \Gamma invariant, we compare the behaviour of the actions of T on \mathrm{Sub}_G and \mathrm{Sub}_\Gamma in terms of distality. Under certain necessary conditions on the Lie group G , we show that T acts distally on \mathrm{Sub}_G if and only if it acts distally on \mathrm{Sub}_\Gamma . We also obtain certain results about the structure of lattices in a connected Lie group.

Upper capacity entropy and packing entropy of saturated sets for amenable group actions

Let $ (X, G) $ be a $ G $-action topological system, where $ G $ is a countable infinite discrete amenable group and $ X $ a compact metric space. In this paper we study the upper capacity entropy and packing entropy for systems with weaker version of specification. We prove that the upper capacity always carries full entropy while there is a variational principle for packing entropy of saturated sets.

Steinberg's fixed point theorem for crystallographic complex reflection groups

Steinberg's fixed point theorem states that given a finite complex reflection group the stabilizer subgroup of a point is generated by reflections that fix this point. This statement is also true for affine Weyl groups. Of the infinite discrete complex reflection groups, it was shown that there are some infinite complex reflections groups that have non-trivial stabilizers that do not contain a single reflection, and therefore, these groups cannot satisfy the fixed point theorem. We thus classify the infinite discrete irreducible complex reflection groups of the infinite family which satisfy the statement of the fixed point theorem.

From affine A4 to affine H2: group-theoretical analysis of fivefold symmetric tilings.

The projections of lattices may be used as models of quasicrystals, and the particular affine extension of the H2 symmetry as a subgroup of A4, discussed in this work, presents a different perspective on fivefold symmetric quasicrystallography. Affine H2 is obtained as the subgroup of affine A4. The infinite discrete group with local dihedral symmetry of order 10 operates on the Coxeter plane of the root and weight lattices of A4 whose Voronoi cells tessellate the 4D Euclidean space possessing the affine A4 symmetry. Facets of the Voronoi cells of the root and weight lattices are identified. Four adjacent rhombohedral facets of the Voronoi cell V(0) of A4 project into the decagonal orbit of H2 as thick and thin rhombuses where long diagonals of the rhombohedra serve as reflection line segments of the reflection operators of H2. It is shown that the thick and thin rhombuses constitute the finite fragments of the tiles of the Coxeter plane with the action of the affine H2 symmetry. Projection of the Voronoi cell of the weight lattice onto the Coxeter plane tessellates the plane with four different tiles: thick and thin rhombuses with different edge lengths obtained from the projection of the square faces and two types of hexagons obtained from the projection of the hexagonal faces of the Voronoi cell. The structure of the local dihedral symmetry H2 fixing a particular point on the Coxeter plane is determined.

Positive entropy implies chaos along any infinite sequence

Let G G be an infinite countable discrete amenable group. For any G G -action on a compact metric space ( X , ρ ) (X,\rho ) , it turns out that if the action has positive topological entropy, then for any sequence { s i } i = 1 + ∞ \{s_i\}_{i=1}^{+\infty } with pairwise distinct elements in G G there exists a Cantor subset K K of X X which is Li–Yorke chaotic along this sequence, that is, for any two distinct points x , y ∈ K x,y\in K , one has \[ lim sup i → + ∞ ρ ( s i x , s i y ) > 0 and lim inf i → + ∞ ρ ( s i x , s i y ) = 0. \limsup _{i\to +\infty }\rho (s_i x,s_iy)>0\ \text {and}\ \liminf _{i\to +\infty }\rho (s_ix,s_iy)=0. \]

Computing Classification of Interacting Fermionic Symmetry-Protected Topological Phases Using Topological Invariants* Supported by the National Natural Science Foundation of China (Grant No. 11874115), the Hong Kong Research Grants Council (Grant No. GRF 14306918), and ANR/RGC Joint Research Scheme (Grant No. A-CUHK402/18).

In recent years, great success has been achieved on the classification of symmetry-protected topological (SPT) phases for interacting fermion systems by using generalized cohomology theory. However, the explicit calculation of generalized cohomology theory is extremely hard due to the difficulty of computing obstruction functions. Based on the physical picture of topological invariants and mathematical techniques in homotopy algebra, we develop an algorithm to resolve this hard problem. It is well known that cochains in the cohomology of the symmetry group, which are used to enumerate the SPT phases, can be expressed equivalently in different linear bases, known as the resolutions. By expressing the cochains in a reduced resolution containing much fewer basis than the choice commonly used in previous studies, the computational cost is drastically reduced. In particular, it reduces the computational cost for infinite discrete symmetry groups, like the wallpaper groups and space groups, from infinity to finity. As examples, we compute the classification of two-dimensional interacting fermionic SPT phases, for all 17 wallpaper symmetry groups.

Packing topological entropy for amenable group actions

AbstractPacking topological entropy is a dynamical analogy of the packing dimension, which can be viewed as a counterpart of Bowen topological entropy. In the present paper we give a systematic study of the packing topological entropy for a continuous G-action dynamical system $(X,G)$ , where X is a compact metric space and G is a countable infinite discrete amenable group. We first prove a variational principle for amenable packing topological entropy: for any Borel subset Z of X, the packing topological entropy of Z equals the supremum of upper local entropy over all Borel probability measures for which the subset Z has full measure. Then we obtain an entropy inequality concerning amenable packing entropy. Finally, we show that the packing topological entropy of the set of generic points for any invariant Borel probability measure $\mu $ coincides with the metric entropy if either $\mu $ is ergodic or the system satisfies a kind of specification property.

𝒵-stability of transformation group C*-algebras

Let ( X , Γ ) (X, \Gamma ) be a free and minimal topological dynamical system, where X X is a separable compact Hausdorff space and Γ \Gamma is a countable infinite discrete amenable group. It is shown that if ( X , Γ ) (X, \Gamma ) has the Uniform Rokhlin Property (URP) and Cuntz comparison of open sets (COS), then m d i m ( X , Γ ) = 0 \mathrm {mdim}(X, \Gamma )=0 implies that ( C ( X ) ⋊ Γ ) ⊗ Z ≅ C ( X ) ⋊ Γ (\mathrm {C}(X) \rtimes \Gamma )\otimes \mathcal Z \cong \mathrm {C}(X) \rtimes \Gamma , where m d i m \mathrm {mdim} is the mean dimension of ( X , Γ ) (X, \Gamma ) , Z \mathcal Z is the Jiang-Su algebra, and C ( X ) ⋊ Γ \mathrm {C}(X) \rtimes \Gamma is the transformation group C*-algebra of ( X , Γ ) (X, \Gamma ) . In particular, in this case, m d i m ( X , Γ ) = 0 \mathrm {mdim}(X, \Gamma )=0 implies that the C*-algebra C ( X ) ⋊ Γ \mathrm {C}(X) \rtimes \Gamma is classified by the Elliott invariant.

Geometric Structures in Group Theory

The conference focused on the use of geometric methods to study infinite groups and the interplay of group theory with other areas. One of the central techniques in geometric group theory is to study infinite discrete groups by their actions on nice, suitable spaces. These spaces often carry an interesting large-scale geometry, such as non-positive curvature or hyperbolicity in the sense of Gromov, or are equipped with rich geometric or combinatorial structure. From these actions one can investigate structural properties of the groups. This connection has become very prominent during the last years. In this context non-discrete topological groups, such as profinite groups or locally compact groups appear quite naturally. Likewise, analytic methods and operator theory play an increasing role in the area.

Conditional variational principles of conditional entropies for amenable group actions * *The work was supported by NSF of China (11671058, 12071049) and 2020CDCGJ019.

Let G be an infinite discrete countable amenable group acting continuously on two compact metrizable spaces X, Y. Assume that φ : (Y, G) → (X, G) is a factor map. Using finite open covers, the conditional topological entropy of φ is defined. The conditional measure-theoretic entropy of φ equals the conditional measure-theoretic entropy of Y to X. With the aid of tiling system of G, the conditional variational principle of φ is studied when (X, G) is an asymptotically h-expansive system. If X = Y and φ is the identity map, the conditional topological entropy of system (X, G) is defined. In the Cartesian square (X × X, G), we define the conditional measure-theoretic entropy of (X, G) to be the defect of the upper semi-continuity of the conditional measure-theoretic entropy of X × X to the first axis. Then the conditional variational principle of (X, G) is obtained.

Bernoulli disjointness

Generalizing a result of Furstenberg, we show that for every infinite discrete group $G$, the Bernoulli flow $2^G$ is disjoint from every minimal $G$-flow. From this, we deduce that the algebra generated by the minimal functions $\mathfrak{A}(G)$ is a proper subalgebra of $\ell^\infty(G)$ and that the enveloping semigroup of the universal minimal flow $M(G)$ is a proper quotient of the universal enveloping semigroup $\beta G$. When $G$ is countable, we also prove that for any metrizable, minimal $G$-flow, there exists a free, minimal flow disjoint from it and that there exist continuum many mutually disjoint minimal, free, metrizable $G$-flows. Finally, improving a result of Frisch, Tamuz, and Vahidi Ferdowsi and answering a question of theirs, we show that if $G$ is a countable icc group, then it admits a free, minimal, proximal flow.

A short proof of Bernoulli disjointness via the local lemma

Recently, Glasner, Tsankov, Weiss, and Zucker showed that if $\Gamma$ is an infinite discrete group, then every minimal $\Gamma$-flow is disjoint from the Bernoulli shift $2^\Gamma$. Their proof is somewhat involved; in particular, it invokes separate arguments for different classes of groups. In this note, we give a short and self-contained proof of their result using purely combinatorial methods applicable to all groups at once. Our proof relies on the Lovasz Local Lemma, an important tool in probabilistic combinatorics that has recently found several applications in the study of dynamical systems.

Sequence entropies and mean sequence dimension for amenable group actions

Let G be an infinite discrete countable amenable group acting continuously on a compact metrizable space X and Ω be a sequence in G. By using open covers of X, a topological invariant, the topological sequence entropy, is defined. It is shown that the topological sequence entropy can also be defined through relative spanning set or relative separate set. We prove that the topological sequence entropy equals the topological entropy multiplying a constant K(Ω) depending only on sequence Ω. The notion of the measure-theoretic sequence entropy is defined. The two sequence entropies are related by the variational principle. The mean topological sequence dimension is defined. It will be nonzero only if the topological sequence entropy is infinite.

Exceptional Algebraic Sets for Infinite Discrete Groups of $$PSL(3,\mathbb {C})$$

In this note we show that the exceptional algebraic set for an infinite discrete group in $$PSL(3,{{\mathbb {C}}})$$ should be a finite union of: complex lines, copies of the Veronese curve or copies of the cubic $$xy^2-z^3$$ .

The Segal conjecture for infinite discrete groups

We formulate and prove a version of the Segal Conjecture for infinite groups. For finite groups it reduces to the original version. The condition that G is finite is replaced in our setting by the assumption that there exists a finite model for the classifying space underline{E}G for proper actions. This assumption is satisfied for instance for word hyperbolic groups or cocompact discrete subgroups of Lie groups with finitely many path components. As a consequence we get for such groups G that the zero-th stable cohomotopy of the classifying space BG is isomorphic to the I-adic completion of the ring given by the zero-th equivariant stable cohomotopy of underline{E}G for I the augmentation ideal.