Volume 28, Issue 5

Since Hol⁡(Fn)$\operatorname*{Hol}(F_{n})$ embeds into Aut⁡(Fn+1)$\operatorname*{Aut}(F_{n+1})$, one can construct inductively the subgroups ℋ(n)${\mathcal{H}}_{(n)}$ of Aut⁡(Fn+1)$\operatorname*{Aut}(F_{n+1})$ by setting ℋ(1)=Hol⁡(F2)${{\mathcal{H}}_{(1)}=\operatorname*{Hol}(F_{2})}$ and ℋ(n)=Fn+1⋊ℋ(n-1)${{\mathcal{H}}_{(n)}=F_{n+1}\rtimes{\mathcal{H}}_{(n-1)}}$. We show that the FJCw holds for ℋ(n)${\mathcal{H}}_{(n)}$. Moreover, we calculate the lower K -theory for the groups ℋ(n)${\mathcal{H}}_{(n)}$.

Let L be a homogeneous divergence form higher order elliptic operator with complex bounded measurable coefficients and (p-⁢(L),p+⁢(L))${(p_{-}(L),p_{+}(L))}$ be the maximal interval of exponents q∈[1,∞]${q\in[1,\infty]}$ such that the semigroup {e-t⁢L}t>0${\{e^{-tL}\}_{t>0}}$ is bounded on Lq⁢(ℝn)${L^{q}(\mathbb{R}^{n})}$. In this article, the authors establish the non-tangential maximal function characterizations of the associated Hardy spaces HLp⁢(ℝn)${H_{L}^{p}(\mathbb{R}^{n})}$ for all p∈(0,p+⁢(L))${p\in(0,p_{+}(L))}$, which when p=1${p=1}$, answers a question asked by Deng, Ding and Yao in [21]. Moreover, the authors characterize HLp⁢(ℝn)${H_{L}^{p}(\mathbb{R}^{n})}$ via various versions of square functions and Lusin-area functions associated to the operator L .

Roughly speaking, the regular subspace of a Dirichlet form is also a regular Dirichlet form on the same state space. It inherits the same form of the original Dirichlet form but possesses a smaller domain. What we are concerned in this paper are the regular subspaces of associated Dirichlet forms of skew product diffusions. A skew product diffusion X is a symmetric Markov process on the product state space E1×E2${E_{1}\times E_{2}}$ and expressed as Xt=(Xt1,XAt2),t≥0,$X_{t}=(X^{1}_{t},X^{2}_{A_{t}}),\quad t\geq 0,$ where Xi${X^{i}}$ is a symmetric diffusion on Ei${E_{i}}$ for i=1,2${i=1,2}$, and A is a positive continuous additive functional of X1${X^{1}}$. One of our main results indicates that any skew product type regular subspace of X , say Yt=(Yt1,YA~t2),t≥0,$Y_{t}=(Y^{1}_{t},{Y^{2}_{\tilde{A}_{t}}}),\quad t\geq 0,$ can be characterized as follows: the associated smooth measure of A~${\tilde{A}}$ is equal to that of A , and Yi${Y^{i}}$ corresponds to a regular subspace of Xi${X^{i}}$ for i=1,2${i=1,2}$. Furthermore, we shall make some discussions on rotationally invariant diffusions on ℝd∖{}${\mathbb{R}^{d}\setminus\{{\mathbf{0}}\}}$, which are special skew product diffusions on (0,∞)×Sd-1${(0,\infty)\times S^{d-1}}$. Our main purpose is to extend a regular subspace of rotationally invariant diffusion on ℝd∖{}${\mathbb{R}^{d}\setminus\{{\mathbf{0}}\}}$ to a new regular Dirichlet form on ℝd${\mathbb{R}^{d}}$. More precisely, fix a regular Dirichlet form (ℰ,ℱ)${(\mathcal{E,F}\kern 0.569055pt)}$ on the state space ℝd${\mathbb{R}^{d}}$. Its part Dirichlet form on ℝd∖{}${\mathbb{R}^{d}\setminus\{{\mathbf{0}}\}}$ is denoted by (ℰ0,ℱ)0${(\mathcal{E}^{0},\mathcal{F}{}^{0})}$. Let (ℰ~0,ℱ~)0${(\tilde{\mathcal{E}}^{0},\tilde{\mathcal{F}}{}^{0})}$ be a regular subspace of (ℰ0,ℱ)0${(\mathcal{E}^{0},\mathcal{F}{}^{0})}$. We want to find a regular subspace (ℰ~,ℱ~)${(\tilde{\mathcal{E}},\tilde{\mathcal{F}}\kern 0.569055pt)}$ of (ℰ,ℱ)${(\mathcal{E,F}\kern 0.569055pt)}$ such that the part Dirichlet form of (ℰ~,ℱ~)${(\tilde{\mathcal{E}},\tilde{\mathcal{F}}\kern 0.569055pt)}$ on ℝd∖{}${\mathbb{R}^{d}\setminus\{{\mathbf{0}}\}}$ is exactly (ℰ~0,ℱ~)0${(\tilde{\mathcal{E}}^{0},\tilde{\mathcal{F}}{}^{0})}$. If (ℰ~,ℱ~)${(\tilde{\mathcal{E}},\tilde{\mathcal{F}}\kern 0.569055pt)}$ exists, we call it a regular extension of (ℰ~0,ℱ~)0${(\tilde{\mathcal{E}}^{0},\tilde{\mathcal{F}}{}^{0})}$. We shall prove that, under a mild assumption, any rotationally invariant type regular subspace of (ℰ0,ℱ)0${(\mathcal{E}^{0},\mathcal{F}{}^{0})}$ has a unique regular extension.

Complex Hermitian random matrices with a unitary symmetry can be distinguished by a weight function. When this is even, it is a known result that the distribution of the singular values can be decomposed as the superposition of two independent eigenvalue sequences distributed according to particular matrix ensembles with chiral unitary symmetry. We give decompositions of the distribution of singular values, and the decimation of the singular values – whereby only even, or odd, labels are observed – for real symmetric random matrices with an orthogonal symmetry, and even weight. This requires further specifying the functional form of the weight to one of three types – Gauss, symmetric Jacobi or Cauchy. Inter-relations between gap probabilities with orthogonal and unitary symmetry follow as a corollary. The Gauss case has appeared in a recent work of Bornemann and La Croix. The Cauchy case, when appropriately specialised and upon stereographic projection, gives decompositions for the analogue of the singular values for the circular unitary and circular orthogonal ensembles.

We study the space of vector valued theta functions for the Weil representation of a positive definite even lattice of rank two with fundamental discriminant. We work out the relation of this space to the corresponding scalar valued theta functions of weight one and determine an orthogonal basis with respect to the Petersson inner product. Moreover, we give an explicit formula for the Petersson norms of the elements of this basis.

We consider generalisations of Thompson’s group V , denoted by Vr⁢(Σ)${V_{r}(\Sigma)}$, which also include the groups of Higman, Stein and Brin. We show that, under some mild hypotheses, Vr⁢(Σ)${V_{r}(\Sigma)}$ is the full automorphism group of a Cantor algebra. Under some further minor restrictions, we prove that these groups are of type F∞${\operatorname{F}_{\infty}}$ and that this implies that also centralisers of finite subgroups are of type F∞${\operatorname{F}_{\infty}}$.

We introduce the class of right annelidan rings, defined by the property that any annihilator right ideal of the ring is comparable with every right ideal of the ring. This class is a common generalization of the classes of domains and right uniserial rings. We obtain results on the structure of right annelidan rings; in particular, we show that all right annelidan rings are Armendariz. We study the relationships between right annelidan rings, chain conditions, and 2-primal rings. For the class of right annelidan rings, we prove a version of the Hopkins–Levitzki Theorem for principal right ideals. We characterize right annelidan group algebras, obtaining a classification that is complete if the zero-divisor problem has a positive solution.

Using S1${S^{1}}$-equivariant symplectic homology, in particular its mean Euler characteristic, of the natural filling of links of Brieskorn–Pham polynomials, we prove the existence of infinitely many inequivalent contact structures on various manifolds, including in dimension 5 the k -fold connected sums of S2×S3${S^{2}\times S^{3}}$ and certain rational homology spheres. We then apply our result to show that on these manifolds the moduli space of classes of positive Sasakian structures has infinitely many components. We also apply our results to give lower bounds on the number of components of the moduli space of Sasaki–Einstein metrics on certain homotopy spheres. Finally, a new family of Sasaki–Einstein metrics of real dimension 20 on S5${S^{5}}$ is exhibited.

We prove that adjoint orbits of semisimple Lie algebras have the structure of symplectic Lefschetz fibrations. We describe the topology of the regular and singular fibres, in particular we calculate their middle Betti numbers.

For Γ a cocompact or cofinite Fuchsian group, we study the hyperbolic lattice point problem in conjugacy classes, which is a modification of the classical hyperbolic lattice point problem. We use large sieve inequalities for the Riemann surfaces Γ\ℍ${{\Gamma\backslash{\mathbb{H}}}}$ to obtain average results for the error term, which are conjecturally optimal. We give a new proof of the error bound O⁢(X2/3)${O(X^{2/3})}$, due to Good. For SL2⁢(ℤ)${{\mathrm{SL}_{2}({\mathbb{Z}})}}$ we interpret our results in terms of indefinite quadratic forms.

In this note I fix a mistake in my previous paper [5]: namely, the result concerning the uniqueness (up to isomorphisms) of such supergroups needs a new formulation and proof. By the same occasion, I explain more in detail the existence result which comes out of the construction of Chevalley supergroups.