Effective minimal subflows of Bernoulli flows

Infinite discrete symmetry group for the Yang-Baxter equations: Spin models

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.

Let $G$ be an infinite discrete group. A classifying space for proper actions\nof $G$ is a proper $G$-CW-complex $X$ such that the fixed point sets $X^H$ are\ncontractible for all finite subgroups $H$ of $G$. In this paper we consider the\nstable analogue of the classifying space for proper actions in the category of\nproper $G$-spectra and study its finiteness properties. We investigate when $G$\nadmits a stable classifying space for proper actions that is finite or of\nfinite type and relate these conditions to the compactness of the sphere\nspectrum in the homotopy category of proper $G$-spectra and to classical\nfiniteness properties of the Weyl groups of finite subgroups of $G$. Finally,\nif the group $G$ is virtually torsion-free we also show that the smallest\npossible dimension of a stable classifying space for proper actions coincides\nwith the virtual cohomological dimension of $G$, thus providing the first\ngeometric interpretation of the virtual cohomological dimension of a group.\n

Expander graphs are ingredients for making concentrating, switching, and sorting networks, and are closely related to sparse, doubly-stochastic matrices called diffusers. The best explicit examples of diffusers are defined by means of the action of elements of the matrix group $SL (2,{\bf Z} )$ on certain finite mathematical objects. Some corresponding, explicit expanders were introduced by Margulis. However, Gabber and Galil were the first to obtain good estimates for the expanders and produce from them a family of directed acyclic superconcentrators having density 271.8. In this paper we review various techniques for making expanders from diffusers. We also demonstrate asymptotic upper bounds on the strength of algebraically defined classes of degree k diffusers. Each upper bound is given as the norm of a diffusion operator on an infinite discrete group, and bounds for several examples are calculated. Numerical evidence is supplied in support of our conjecture that these bounds can be achieved by certain algebraically defined examples. The conjecture, if true, would lead to superconcentrators of density less than 58.

The purpose of this paper is to investigate the topological and analytical restrictions on a domain D in euclidean n-space R n on which an infinite discrete quasiconformal group can act. We will see that the restrictions are indeed severe, unlike the case of a discrete group of topological or differentiable homeomorphisms.KeywordsUnit BallLimit PointUniversal CoverTangent PlaneQuasiconformal MappingThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Given an infinite countable discrete amenable group Γ \Gamma , we construct explicitly sharply weak mixing nonsingular Poisson Γ \Gamma -actions of each Krieger’s type: I I I λ III_\lambda , for λ ∈ [ 0 , 1 ] \lambda \in [0,1] , and I I ∞ II_\infty . The result is new even for Γ = Z \Gamma =\mathbb {Z} . As these Poisson suspension actions are over very special dissipative base, we obtain also new examples of sharply weak mixing nonsingular Bernoulli Γ \Gamma -actions and infinite direct product of finite type systems of each possible Krieger’s type.

Self-dual models with theta terms

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.

A connection between integrability properties and general statistical properties of the spectra of symmetric transfer matrices of the asymmetric eight-vertex model is studied using random matrix theory (eigenvalue spacing distribution and spectral rigidity). For Yang-Baxter integrable cases, including free-fermion solutions, we have found a Poissonian behavior, whereas level repulsion close to the Wigner distribution is found for non-integrable models. For the asymmetric eight-vertex model, however, the level repulsion can also disappearand the Poisson distribution be recovered on (non Yang--Baxter integrable) algebraic varieties, the so-called disorder varieties. We also present an infinite set of algebraic varieties which are stable under the action of an infinite discrete symmetry group of the parameter space. These varieties are possible loci for free parafermions. Using our numerical criterion we have tested the generic calculability of the model on these algebraic varieties.

One-dimensional flows in the quantum Hall system

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.

There is a large class of Gaussian lattice models which are self-dual with respect to inversion of coupling constants. When a theta term is added, this duality may extend to an invariance under an action of an infinite discrete modular group on the coupling parameter space. In particular, 4-dimensional Abelian lattice gauge theories possess a Sp(2k, Z) modular symmetry, and string compactifications on d-dimensional tori are invariant under SO(16 + d, d; Z). We review these results and their applications, and discuss the classification of self-dual string compactifications.KeywordsGauge TheoryPartition FunctionModulus SpaceFundamental DomainHeterotic StringThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

For a generally infinite noncommutative discrete group G, we study derivation algebras in the group algebra of G in terms of characters on a groupoid associated with the group. We obtain necessary conditions for a character to define a derivation. Using these conditions, we analyze some examples. In particular, we describe a derivation algebra in the case when the group is a nilpotent group of rank 2.

We construct, for each convex polytope, possibly nonrational and nonsimple, a family of compact spaces that are stratified by quasifolds, i.e. each of these spaces is a collection of quasifolds glued together in an suitable way. A quasifold is a space locally modelled on \({\mathbb{R}^{k}}\) modulo the action of a discrete, possibly infinite, group. The way strata are glued to each other also involves the action of an (infinite) discrete group. Each stratified space is endowed with a symplectic structure and a moment mapping having the property that its image gives the original polytope back. These spaces may be viewed as a natural generalization of symplectic toric varieties to the nonrational setting.

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.