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In this paper we study the spaces of non-compact real algebraic curves, i.e. pairs $(P,\tau)$, where $P$ is a compact Riemann surface with a finite number of holes and punctures and $\tau:P\to P$ is an anti-holomorphic involution. We describe the uniformisation of non-compact real algebraic curves by Fuchsian groups. We construct the spaces of non-compact real algebraic curves and describe their connected components. We prove that any connected component is homeomorphic to a quotient of a finite-dimensional real vector space by a discrete group and determine the dimensions of these vector spaces., Comment: 9 pages, 5 figures

Given any faithful 3-transitive action of a countable discrete group G on an infinite set X. Assume either the action is sharply 3-transitive or the stabilizer subgroup of any two distinct points is I.C.C. We show that the stabilizer subgroup of any point in X generates a maximal and rigid von Neumann subalgebra in LG. This extends a previous result by Jiang [4] , who proved the same result but under stronger assumption of 4-transitivity.

Let $G$ be a noncompact semisimple Lie group, $\Gamma$ be an irreducible cocompact lattice in $G$, and $P 0$, there is a $g\in P$ such that $gA$ is $\epsilon$-dense. We also prove a similar result for certain discrete group actions on $\mathbb T^n$.

It is well-known that the existence of more than two ends in the sense of J.R. Stallings for a finitely generated discrete group $G$ can be detected on the cohomology group $\mathrm{H}^1(G,R[G])$, where $R$ is either a finite field, the ring of integers or the field of rational numbers. It will be shown (cf. Theorem A*) that for a compactly generated totally disconnected locally compact group $G$ the same information about the number of ends of $G$ in the sense of H. Abels can be provided by $\mathrm{dH}^1(G,\mathrm{Bi}(G))$, where $\mathrm{Bi}(G)$ is the rational discrete standard bimodule of $G$, and $\mathrm{dH}^\bullet(G,\_)$ denotes rational discrete cohomology as introduced in [6]. As a consequence one has that the class of fundamental groups of a finite graph of profinite groups coincides with the class of compactly presented totally disconnected locally compact groups of rational discrete cohomological dimension at most 1 (cf. Theorem B)., more detailed version - some corrections have been made -title has been changed

We completely characterize connected Lie groups all of whose countable subgroups are weakly amenable. We also provide a characterization of connected semisimple Lie groups that are weakly amenable. Finally, we show that a connected Lie group is weakly amenable if the group is weakly amenable as a discrete group., Comment: Final version. To appear in Pacific Journal of Mathematics

We study the Furstenberg-entropy realization problem for stationary actions. It is shown that for finitely supported probability measures on free groups, any a-priori possible entropy value can be realized as the entropy of an ergodic stationary action. This generalizes results of Bowen. The stationary actions we construct arise via invariant random subgroups (IRSs), based on ideas of Bowen and Kaimanovich. We provide a general framework for constructing a continuum of ergodic IRSs for a discrete group under some algebraic conditions, which gives a continuum of entropy values. Our tools apply for example, for certain extensions of the group of finitely supported permutations and lamplighter groups, hence establishing full realization results for these groups. For the free group, we construct the IRSs via a geometric construction of subgroups, by describing their Schreier graphs. The analysis of the entropy of these spaces is obtained by studying the random walk on the appropriate Schreier graphs., Comment: 31 pages

We propose a one-loop induced radiative seesaw model applying a modular $S_3$ flavor symmetry, which is known as the minimal non-Abelian discrete group. In this scenario, dark matter (DM) candidate is correlated with neutrinos and lepton flavor violations (LFVs). We show several predictions of mixings and phases satisfying LFVs, observed relic density, and neutrino oscillation data., 14 pages, 3 figures, 2 tables: model changed,reference added; version accepted for publication in Physical Review D

We study the K-theory ranks for crossed products of C_-algebras by the group of integers. As an application, we obtain certain estimates for the K-theory ranks of the group C_-algebras of torsion free, finitely generated, nilpotent or solvable discrete groups, written as successive semi-direct products. Resumen Estudiamos los rangos de K-teoría para productos cruzados de C_-álgebras por el grupo de los enteros. Como aplicación, obtenemos ciertas estimaciones para los rangos de K- teoría de las álgebras de grupos libres de torsión, finitamente generados, nilpotentes o solubles, escritos como productos semidirectos sucesivos.

We study the K-theory ranks for crossed products of C_-algebras by the group of integers. As an application, we obtain certain estimates for the K-theory ranks of the group C_-algebras of torsion free, finitely generated, nilpotent or solvable discrete groups, written as successive semi-direct products.

If G is a countable discrete group acting linearly on a finite-dimensional vector space over any topological field, then the groups of coboundaries are closed for the product topology in all degrees, and hence the cohomology is reduced in all degrees. This is deduced from a more general automatic-closure theorem for continuous linear transformations between inverse limits of finite-dimensional vector spaces., Comment: 10 pages [v3:] Small changes following referee report

Let G be a finitely generated group of isometries of $$\mathbb {H}$$ , hyperbolic 2-space. The discreteness problem is to determine whether or not G is discrete. Even in the case of a two generator non-elementary subgroup of $$\mathbb {H}$$ (equivalently $$PSL(2,{\mathbb {R}})$$ ) the problem requires an algorithm (Gilman and Maskit in Mich Math J 38:13–32, 1991; Gilman in two-generator discrete subgroups of $$PSL(2, {\mathbb {R}})$$ , 1995). If G is discrete, one can ask when adjoining an nth root of a generator results in a discrete group. In this paper we first address the issue for pairs of hyperbolic generators in $$PSL(2, \mathbb {R})$$ with disjoint axes and obtain necessary and sufficient conditions for adjoining roots for the case when the two hyperbolics have a hyperbolic product and are what as known as stopping generators for the Gilman–Maskit algorithm (Gilman and Maskit 1991; Gilman and Keen in Cutting sequences and palindromes in geometry of Riemann surfaces, 2009). Stopping generators are generators that correspond to certain generalized Coxeter triangle group. We give an algorithmic solution in the other cases that are not stopping generators. Our solution applies to all other types of pairs of stopping generators that arise in what is known as the intertwining case. The results are geometrically motivated and stated as such, but also can be given computationally using the corresponding matrices.

Hardy’s uncertainty principle for the Gabor transform is proved for locally compact abelian groups having noncompact identity component and groups of the form$\mathbb{R}^{n}\times K$, where$K$is a compact group having irreducible representations of bounded dimension. We also show that Hardy’s theorem fails for a connected nilpotent Lie group$G$which admits a square integrable irreducible representation. Further, a similar conclusion is made for groups of the form$G\times D$, where$D$is a discrete group.

Let G be a discrete group with Property (T). It is a standard fact that, in a unitary representation of G on a Hilbert space ℋ {\mathcal{H}} , almost invariant vectors are close to invariant vectors, in a quantitative way. We begin by showing that, if a unitary representation has some vector whose coefficient function is close to a coefficient function of some finite-dimensional unitary representation σ, then the vector is close to a sub-representation isomorphic to σ: this makes quantitative a result of P. S. Wang. We use that to give a new proof of a result by D. Kerr, H. Li and M. Pichot, that a group G with Property (T) and such that C * ⁢ ( G ) {C^{*}(G)} is residually finite-dimensional, admits a unitary representation which is generic (i.e. the orbit of this representation in R ⁢ e ⁢ p ⁢ ( G , ℋ ) {Rep(G,\mathcal{H})} under the unitary group U ⁢ ( ℋ ) {U(\mathcal{H})} is comeager). We also show that, under the same assumptions, the set of representations equivalent to a Koopman representation is comeager in Rep ⁢ ( G , ℋ ) {\mathrm{Rep}(G,\mathcal{H})} .

We study the uniform Roe algebras associated to locally finite groups. We show that for two countable locally finite groups $\Gamma$ and $\Lambda$, the associated uniform Roe algebras $C^*_u(\Gamma)$ and $C^*_u(\Lambda)$ are $*$-isomorphic if and only if their $K_0$ groups are isomorphic as ordered abelian groups with units. This can be seen as a non-separable non-simple analogue of the Glimm-Elliott classification of UHF algebras. To the best of our knowledge, this is the first classification result for a class of non-separable unital $C^*$-algebras. Along the way we also obtain a rigidity result: two countable locally finite groups are bijectively coarsely equivalent if and only if the associated uniform Roe algebras are $*$-isomorphic. Finally, we give a summary of $C^*$-algebraic characterizations for (not necessarily countable) locally finite discrete groups in terms of their uniform Roe algebras. In particular, we show that a discrete group $\Gamma$ is locally finite if and only if the associated uniform Roe algebra $\ell^\infty(\Gamma)\rtimes_r \Gamma$ is locally finite-dimensional., Comment: 20 pages

A generalization of Mallats classical multiresolution analysis, based on thetheory of spectral pairs, was considered in two articles by Gabardo and Nashed. In thissetting, the associated translation set is no longer a discrete subgroup of R but a spectrumassociated with a certain one-dimensional spectral pair and the associated dilation is aneven positive integer related to the given spectral pair. In this paper, we construct dualwavelets which are associated with Nonuniform Multiresolution Analysis. We show thatif the translates of the scaling functions of two multiresolution analyses are biorthogonal,then the associated wavelet families are also biorthogonal. Under mild assumptions onthe scaling functions and the wavelets, we also show that the wavelets generate Rieszbases

A sequence in a $$C^*$$ -algebra A is called completely Sidon if its span in A is completely isomorphic to the operator space version of the space $$\ell _1$$ (i.e. $$\ell _1$$ equipped with its maximal operator space structure). The latter can also be described as the span of the free unitary generators in the (full) $$C^*$$ -algebra of the free group $$\mathbb {F}_\infty $$ with countably infinitely many generators. Our main result is a generalization to this context of Drury’s classical theorem stating that Sidon sets are stable under finite unions. In the particular case when $$A=C^*(G)$$ the (maximal) $$C^*$$ -algebra of a discrete group G, we recover the non-commutative (operator space) version of Drury’s theorem that we recently proved. We also give several non-commutative generalizations of our recent work on uniformly bounded orthonormal systems to the case of von Neumann algebras equipped with normal faithful tracial states.

Chung and Jiang showed that, if a one ended group contains an infinite order element, then every continuous cocycle over the full shift on that group, taking values in a discrete group, must be cohomologous to a homomorphism. We show that their conclusion holds for all one ended groups, so that the hypothesis of admitting an infinite order element may be omitted., Comment: 9 pages, 4 figures, terminology has been standardized, and some typos have been corrected

For a discrete group G, we use the natural correspondence between ideals in the Boolean algebra P G of subsets of G and closed subsets in the Stone–Cech compactification βG as a right topological semigroup to introduce and characterize some new ideals in βG. We show that if a group G is either countable or Abelian then βG contains no ideals that are maximal among the closed proper subideals of G ⁎ , G ⁎ = β G ∖ G , but this statement does not hold for the group S κ of all permutations of an infinite cardinal κ. We characterize the minimal closed ideal in βG containing all idempotents of G ⁎ .

We consider the traces on submanifolds of G-operators generated by pseudodifferential operators and operators of shift along the orbits of a discrete group G. Such operators arise in various problems in differential equations and mathematical physics, for example, in Sobolev problems. We show that the trace of a G-operator on a submanifold is the sum of a pseudodifferential operator on the submanifold and a G-operator concentrated on a sub-submanifold.

We study index theory for manifolds with Baas-Sullivan singularities using geometric K-homology with coefficients in a unital C*-algebra. In particular, we define a natural analog of the Baum-Connes assembly map for a torsion-free discrete group in the context of these singular spaces. The cases of singularities modelled on k-points (i.e., z/k-manifolds) and the circle are discussed in detail. In the case of the former, the associated index theorem is related to the Freed-Melrose index theorem; in the case of latter, the index theorem is related to work of Rosenberg.

A discrete group G is amenable if there exists a finitely additive probability measure on G which is invariant under left translations and is defined on all subsets of G. It is proved that ...

The Hardy-Littlewood inequality on $\mathbb{T}$ compares the $L^p$-norm of a function with a weighted $\ell^p$-norm of its Fourier coefficients. The approach has recently been studied for compact homogeneous spaces and we study a natural analogue in the framework of compact quantum groups. Especially, in the case of the reduced group $C^*$-algebras and free quantum groups, we establish explicit $L^p-\ell^p$ inequalities through inherent information of underlying quantum group, such as growth rate and rapid decay property. Moreover, we show sharpness of the inequalities in a large class, including $C(G)$ with compact Lie group, $C_r^*(G)$ with polynomially growing discrete group and free quantum groups $O_N^+$, $S_N^+$., 22 pages

Given the large class of groups already known to be sofic, there is seemingly a shortfall in results concerning their permanence properties. We address this problem for wreath products, and in particular investigate the behaviour of more general metric approximations of groups under wreath products. Our main result is the following. Suppose that $H$ is a sofic group and $G$ is a countable, discrete group. If $G$ is sofic, hyperlinear, weakly sofic, or linear sofic, then $G\wr H$ is also sofic, hyperlinear, weakly sofic, or linear sofic respectively. In each case we construct relevant metric approximations, extending a general construction of metric approximations for $G\wr H$ that uses soficity of $H$.

We consider the nilpotent left-invariant sub-Riemannian structure on the Engel group. This structure gives a fundamental local approximation of a generic rank 2 sub-Riemannian structure on a 4-manifold near a generic point (in particular, of the kinematic models of a car with a trailer). On the other hand, this is the simplest sub-Riemannian structure of step three. We describe the global structure of the cut locus (the set of points where geodesics lose their global optimality), the Maxwell set (the set of points that admit more than one minimizer), and the intersection of the cut locus with the caustic (the set of conjugate points along all geodesics). The group of symmetries of the cut locus is described: it is generated by a one-parameter group of dilations R+ and a discrete group of reflections Z2 × Z2 × Z2. The cut locus admits a stratification with 6 three-dimensional strata, 12 two-dimensional strata, and 2 one-dimensional strata. Three-dimensional strata of the cut locus are Maxwell strata of multiplicity 2 (for each point there are 2 minimizers). Two-dimensional strata of the cut locus consist of conjugate points. Finally, one-dimensional strata are Maxwell strata of infinite multiplicity, they consist of conjugate points as well. Projections of sub-Riemannian geodesics to the 2-dimensional plane of the distribution are Euler elasticae. For each point of the cut locus, we describe the Euler elasticae corresponding to minimizers coming to this point. Finally, we describe the structure of the optimal synthesis, i. e., the set of minimizers for each terminal point in the Engel group.

Let π : ( T , X ) → ( T , Z ) be an extension of flows with phase group T. A point x in X is π-distal if x is proximal to at most itself in π − 1 π x ∩ T x ‾ under ( T , X ) . We study the π-distal points using combinatorial methods. We present characterizations of π-distal points using product IP/ C w /C-recurrence, dynamics syndetic sets, distal sets, IP⁎-sets, and C⁎-sets in T. Moreover, we give the dynamics realization of IP-set of any discrete group by IP-recurrent point and the dynamics realization of C-set of Z d by C-recurrent point. The IP ⁎ -recurrence of π-distal points introduced here is useful for simplifying the proof of Furstenberg's structure theorem.

We show how to construct a graded locally compact Hausdorff \'etale groupoid from a C*-algebra carrying a coaction of a discrete group, together with a suitable abelian subalgebra. We call this groupoid the extended Weyl groupoid. When the coaction is trivial and the subalgebra is Cartan, our groupoid agrees with Renault's Weyl groupoid. We prove that if G is a second-countable locally compact \'etale groupoid carrying a grading of a discrete group, and if the interior of the trivially graded isotropy is abelian and torsion free, then the extended Weyl groupoid of its reduced C*-algebra is isomorphic as a graded groupoid to G. In particular, two such groupoids are isomorphic as graded groupoids if and only if there is an equivariant diagonal-preserving isomorphism of their reduced C*-algebras. We introduce graded equivalence of groupoids, and establish that two graded groupoids in which the trivially graded isotropy has torsion-free abelian interior are graded equivalent if and only if there is an equivariant diagonal-preserving Morita equivalence between their reduced C*-algebras. We use these results to establish rigidity results for a number of classes of dynamical systems, including all actions of the natural numbers by local homeomorphisms of locally compact Hausdorff spaces., Comment: v2: 45 pages; removed notion of "weakly Cartan subalgebra" as it is equivalent to a sigma-unital abelian subalgebra (see Lemma 4.1) and updated numerous statements accordingly. Thanks to R. Meyer and B. Kwasniewski for pointing this out. Numerous other minor corrections and improvements. This version to appear in Advances in Mathematics

Let Γ be some discrete subgroup of SO°(n + 1, R) with finite Bowen-Margulis-Sullivan measure. We study the dynamics of the Bowen-Margulis-Sullivan measure with respect to closed connected subspaces of the N component in some Iwasawa decomposition SO°(n+1, R) = KAN. We also study the dimension of projected Patterson-Sullivan measures along some fixed direction.

Let $\Gamma$ be a countable discrete group, and let $\pi\colon \Gamma\to {\rm{GL}}(H)$ be a representation of $\Gamma$ by invertible operators on a separable Hilbert space $H$. We show that the semidirect product group $G=H\rtimes_{\pi}\Gamma$ is SIN ($G$ admits a two-sided invariant metric compatible with its topology) and unitarily representable ($G$ embeds into the unitary group $\mathcal{U}(\ell^2(\mathbb N))$), if and only if $\pi$ is uniformly bounded, and that $\pi$ is unitarizable if and only if $G$ is of finite type: that is, $G$ embeds into the unitary group of a II$_1$-factor. Consequently, we show that a unitarily representable Polish SIN groups need not be of finite type, answering a question of Sorin Popa. The key point in our argument is an equivariant version of the Maurey--Nikishin factorization theorem for continuous maps from a Hilbert space to the space $L^0(X,m)$ of all measurable maps on a probability space., Comment: 27 pages v2 minor changes: corrected the hypothesis in Corollary 3.16 and first few lines of the proof of Lemma 3.7

We consider the cohomology group H-1(Gamma, p) of a discrete subgroup Gamma subset of G = SU(n, 1) and the symmetric tensor representation p on S-m(c(n+1)). We give an elementary proof of the Eichler-Shimura isomorphism that harmonic forms H-1(Gamma\ G/K, p) are (0, 1)-forms for the automorphic holomorphic bundle induced by the representation S-m(C-n) of K.

We define and study the weak Haagerup property for $C^{*}$ -algebras in this article. A $C^{*}$ -algebra with the Haagerup property always has the weak Haagerup property. We prove that a discrete group has the weak Haagerup property if and only if its reduced group $C^{*}$ -algebra also has that property. Moreover, we consider the permanence of the weak Haagerup property under a few canonical constructions of $C^{*}$ -algebras.

If G is a discrete group with bounded torsion, the Strong Atiyah Conjecture predicts that the $$L^2$$ -Betti numbers of any G-space are rational, with denominators determined by the orders of the torsion subgroups. We prove the conjecture for some new examples of groups, including the class of virtually sparse special groups. We then show that the conjecture can be used to compute the $$L^2$$ -cohomology of certain Coxeter groups that have highly symmetric nerves.

Applying methods of CR-geometry, we give a solution to the local equivalence problem for second order (smooth or analytic) ordinary differential equations. We do so by presenting a complete normal form (which is smooth or analytic respectively) for this class of ordinary differential equations (ODEs). The normal form is optimal in the sense that it is defined up to the automorphism group of the model (flat) ODE y ″ = 0. For a generic ODE, we also provide a unique (up to a discrete group action) normal form. By doing so, we give a solution to a problem which remained unsolved since the work of Arnold (1988). As another application of the normal form, we obtain distinguished curves associated with a differential equation that we call chains due to their analogy with the chains defined by Chern and Moser (Acta Math. 7;133:219–271).

Let G be a finite group and σ = {σ i |i∈I} be a partition of the set of all primes P. A set H of subgroups of G is said to be a complete Hall σ-set of G if every non-identity member of H is a Hall σ i -subgroup of G and H contains exactly one Hall σ i -subgroup of G for every σ i ∈ σ(G). A subgroup H is said to be σ-permutable if G possesses a complete Hall σ-set H such that HA x = A x H for all A ∈ H and all x ∈ G. Let H be a subgroup of G. Then we say that: (1) H is σ-embedded in G if there exists a σ-permutable subgroup T of G such that HT = H σG and H ∩ T ≤ H σG , where H σG is the subgroup of H generated by all those subgroups of H which are σ-permutable in G, and H σG is the σ-permutable closure of H, that is, the intersection of all σ-permutable subgroups of G containing H. (2) H is σ-n-embedded in G if there exists a normal subgroup T of G such that HT = H G and H ∩ T ≤ H σG . In this paper, we study the properties of the new embedding subgroups and use them to determine the structure of finite groups.

Jacobi-like forms for a discrete subgroup $$\Gamma $$ of $$SL(2, \mathbb R)$$ are formal power series which generalize Jacobi forms, and they are in one-to-one correspondence with automorphic pseudodifferential operators for $$\Gamma $$ . The well-known Cohen–Kuznetsov lifting of a modular form f provides a Jacobi-like form and therefore an automorphic pseudodifferential operator associated to f. Given a pair $$(\lambda , \mu )$$ of integers, automorphic pseudodifferential operators can be extended to those of mixed weight. We show that each coefficient of an automorphic pseudodifferential operator of mixed weight is a quasimodular form and prove the existence of a lifting of Cohen–Kuznetsov type for each quasimodular form.