Finite groups all of whose second maximal subgroups are {\mathcal{H }}_p-groups

Let $G$ be a finite group. A subgroup $H$ of $G$ is called an $\mathcal{H }$ -subgroup of $G$ if $N_G(H)\cap H^g\le H$ for all $g\in G$ . A group $G$ is said to be an ${\mathcal{H }}_p$ -group if every cyclic subgroup of $G$ of prime order or order 4 is an $\mathcal{H }$ -subgroup of $G$ . In this paper, the structure of a finite group all of whose second maximal subgroups are ${\mathcal{H }}_p$ -subgroups has been characterized.     

Growth estimates for orbits of self-adjoint groups

Let $G$ denote a closed, connected, self-adjoint, noncompact subgroup of $GL(n,\mathbb R )$ , and let $d_{R}$ and $d_{L}$ denote respectively the right and left invariant Riemannian metrics defined by the canonical inner product on $M(n,\mathbb R ) = T_{I} GL(n,\mathbb R )$ . Let $v$ be a nonzero vector of $\mathbb R ^{n}$ such that the orbit $G(v)$ is unbounded in $\mathbb R ^{n}$ . Then the function $g \rightarrow d_{R}(g, G_{v})$ is unbounded, where $G_{v} = \{g \in G : g(v) = v \}$ , and we obtain algebraically defined upper and lower bounds $\lambda ^{+}(v)$ and $\lambda ^{-}(v)$ for the asymptotic behavior of the function $\frac{log|g(v)|}{d_{R}(g, G_{v})}$ as $d_{R}(g, G_{v}) \rightarrow \infty $ . The upper bound $\lambda ^{+}(v)$ is at most 1. The orbit $G(v)$ is closed in $\mathbb R ^{n} \Leftrightarrow \lambda ^{-}(w)$ is positive for some w $\in G(v)$ . If $G_{v}$ is compact, then $g \rightarrow |d_{R}(g,I) - d_{L}(g,I)|$ is uniformly bounded in $G$ , and the exponents $\lambda ^{+}(v)$ and $\lambda ^{-}(v)$ are sharp upper and lower asymptotic bounds for the functions $\frac{log|g(v)|}{d_{R}(g,I)}$ and $\frac{log|g(v)|}{d_{L}(g,I)}$ as $d_{R}(g,I) \rightarrow \infty $ or as $d_{L}(g,I) \rightarrow \infty $ . However, we show by example that if $G_{v}$ is noncompact, then there need not exist asymptotic upper and lower bounds for the function $\frac{log|g(v)|}{d_{L}(g, G_{v})}$ as $d_{L}(g, G_{v}) \rightarrow \infty $ . The results apply to representations of noncompact semisimple Lie groups $G$ on finite dimensional real vector spaces. We compute $\lambda ^{+}$ and $\lambda ^{-}$ for the irreducible, real representations of $SL(2,\mathbb R )$ , and we show that if the dimension of the $SL(2,\mathbb R )$ -module $V$ is odd, then $\lambda ^{+} = \lambda ^{-}$ on a nonempty open subset of $V$ . We show that the function $\lambda ^{-}$ is $K$ -invariant, where $K = O(n,\mathbb R ) \cap G$ . We do not know if $\lambda ^{-}$ is $G$ -invariant.     

Wild quotient singularities of surfaces

Let $(B,\mathcal{M }_B)$ be a noetherian regular local ring of dimension $2$ with residue field $B/\mathcal{M }_B$ of characteristic $p>0$ . Assume that $B$ is endowed with an action of a finite cyclic group $H$ whose order is divisible by $p$ . Associated with a resolution of singularities of $\mathrm{Spec}B^H$ is a resolution graph $G$ and an intersection matrix $N$ . We prove in this article three structural properties of wild quotient singularities, which suggest that in general, one should expect when $H= \mathbb{Z }/p\mathbb{Z }$ that the graph $G$ is a tree, that the Smith group $\mathbb{Z }^n/\mathrm{Im}(N)$ is killed by $p$ , and that the fundamental cycle $Z$ has self-intersection $|Z^2|\le p$ . We undertake a combinatorial study of intersection matrices $N$ with a view towards the explicit determination of the invariants $\mathbb{Z }^n/\mathrm{Im}(N)$ and $Z$ . We also exhibit explicitly the resolution graphs of an infinite set of wild $\mathbb{Z }/2\mathbb{Z }$ -singularities, using some results on elliptic curves with potentially good ordinary reduction which could be of independent interest.     

Modular categories associated with unipotent groups

Let $G$ be a unipotent algebraic group over an algebraically closed field $\mathtt{k }$ of characteristic $p>0$ and let $l\ne p$ be another prime. Let $e$ be a minimal idempotent in $\mathcal{D }_G(G)$ , the $\overline{\mathbb{Q }}_l$ -linear triangulated braided monoidal category of $G$ -equivariant (for the conjugation action) $\overline{\mathbb{Q }}_l$ -complexes on $G$ under convolution (with compact support) of complexes. Then, by a construction due to Boyarchenko and Drinfeld, we can associate to $G$ and $e$ a modular category $\mathcal{M }_{G,e}$ . In this paper, we prove that the modular categories that arise in this way from unipotent groups are precisely those in the class $\mathfrak{C }_p^{\pm }$ .     

Normality and smoothness of simple linear group compactifications

Given a semisimple algebraic group $G$ , we characterize the normality and the smoothness of its simple linear compactifications, namely those equivariant $G\times G$ -compactifications possessing a unique closed orbit which arise in a projective space of the shape $\mathbb{P }(\mathrm{End}(V))$ , where $V$ is a finite dimensional rational $G$ -module. Both the characterizations are purely combinatorial and are expressed in terms of the highest weights of $V$ . In particular, we show that ${\mathrm{Sp}}(2r)$ (with $r \geqslant 1$ ) is the unique non-adjoint simple group which admits a simple smooth compactification.     

On generalizing transitivity, persistence, and sensitivity

A subgroup property $\alpha $ is transitive in a group $G$ if $U \alpha V$ and $V \alpha G$ imply that $U \alpha G$ whenever $U \le V \le G$ , and $\alpha $ is persistent in $G$ if $U \alpha G$ implies that $U \alpha V$ whenever $U \le V \le G$ . Even though a subgroup property $\alpha $ may be neither transitive nor persistent, a given subgroup $U$ may have the property that each $\alpha $ -subgroup of $U$ is an $\alpha $ -subgroup of $G$ , or that each $\alpha $ -subgroup of $G$ in $U$ is an $\alpha $ -subgroup of $U$ . We call these subgroup properties $\alpha $ -transitivity and $\alpha $ -persistence, respectively. We introduce and develop the notions of $\alpha $ -transitivity and $\alpha $ -persistence, and we establish how the former property is related to $\alpha $ -sensitivity. In order to demonstrate how these concepts can be used, we apply the results to the cases in which $\alpha $ is replaced with “normal” and the “cover-avoidance property.” We also suggest ways in which the theory can be developed further.     

A ramification theorem for the ratio of canonical forms of flat surfaces in hyperbolic three-space

A group $G$ is said to be periodic if for every $g\in G$ there exists a positive integer $n$ with $g^n=\mathrm{Id}$ . We prove that a finitely generated periodic group of homeomorphisms on the 2-torus that preserves a probability measure $\mu $ is finite. Moreover if the group consists of homeomorphisms isotopic to the identity, then it is abelian and acts freely on $\mathbb{T }^2$ . In the Appendix, we show that every finitely generated 2-group of toral homeomorphisms is finite.     

On minimal non-M_rC-groups

A group $G$ is said to be a minimax group if it has a finite series whose factors satisfy either the minimal or the maximal condition. Let $D(G)$ denotes the subgroup of $G$ generated by all the Chernikov divisible normal subgroups of $G$ . If $G$ is a soluble-by-finite minimax group and if $D(G)=1$ , then $G$ is said to be a reduced minimax group. Also $G$ is said to be an $ M_{r}C$ -group (respectively, $PC$ -group), if $G/C_{G} \left(x^{G}\right)$ is a reduced minimax (respectively, polycyclic-by-finite) group for all $x\in G$ . These are generalisations of the familiar property of being an $FC$ -group. Finally, if $\mathfrak X $ is a class of groups, then $G$ is said to be a minimal non- $\mathfrak X $ -group if it is not an $\mathfrak X $ -group but all of whose proper subgroups are $\mathfrak X $ -groups. Belyaev and Sesekin characterized minimal non- $FC$ -groups when they have a non-trivial finite or abelian factor group. Here we prove that if $G$ is a group that has a proper subgroup of finite index, then $G$ is a minimal non- $M_{r}C$ -group (respectively, non- $PC$ -group) if, and only if, $G$ is a minimal non- $FC$ -group.     

On locally finite groups with a four-subgroup whose centralizer is small

Let $G$ be a locally finite group which contains a non-cyclic subgroup $V$ of order four such that $C_{G}\left( V\right) $ is finite and $C_{G}\left( \phi \right)$ has finite exponent for some $\phi \in V$ . We show that $[G,\phi ]^{\prime }$ has finite exponent. This enables us to deduce that $G$ has a normal series $1\le G_1\le G_2\le G_3\le G$ such that $G_1$ and $G/G_2$ have finite exponents while $G_2/G_1$ is abelian. Moreover $G_3$ is hyperabelian and has finite index in $G$ .     

Compactification de Chabauty de l’espace des sous-groupes de Cartan de {\text{ SL}}_n(\mathbb{R })

Let $G$ be a real semisimple Lie group with finite center, with a finite number of connected components and without compact factor. We are interested in the homogeneous space of Cartan subgroups of $G$ , which can be also seen as the space of maximal flats of the symmetric space of $G$ . We define its Chabauty compactification as the closure in the space of closed subgroups of $G$ , endowed with the Chabauty topology. We show that when the real rank of $G$ is 1, or when $G={\text{ SL}}_3(\mathbb{R })$ or ${\text{ SL}}_4(\mathbb{R })$ , this compactification is the set of all closed connected abelian subgroups of dimension the real rank of $G$ , with real spectrum. And in the case of ${\text{ SL}}_3(\mathbb{R })$ , we study its topology more closely and we show that it is simply connected.     

Character varieties of abelian groups

We prove that for every reductive group $G$ with a maximal torus ${\mathbb {T}}$ and the Weyl group $W,\, {\mathbb {T}}^N/W$ is the normalization of the irreducible component, $X_G^0({\mathbb {Z}}^N)$ , of the $G$ -character variety $X_G({\mathbb {Z}}^N)$ of ${\mathbb {Z}}^N$ containing the trivial representation. We also prove that $X_G^0({\mathbb {Z}}^N)={\mathbb {T}}^N/W$ for all classical groups. Additionally, we prove that even though there are no irreducible representations in $X_G^0({\mathbb {Z}}^N)$ for non-abelian $G$ , the tangent spaces to $X_G^0({\mathbb {Z}}^N)$ coincide with $H^1({\mathbb {Z}}^N, Ad\, \rho )$ . Consequently, $X_G^0({\mathbb {Z}}^2)$ , has the “Goldman” symplectic form for which the combinatorial formulas for Goldman bracket hold.     

Sufficient conditions for Willmore immersions in \mathbb{R }^3 to be minimal surfaces

We provide two sharp sufficient conditions for immersed Willmore surfaces in $\mathbb{R }^3$ to be already minimal surfaces, i.e. to have vanishing mean curvature on their entire domains. These results turn out to be particularly suitable for applications to Willmore graphs. We can therefore show that Willmore graphs on bounded $C^4$ -domains $\overline{\varOmega }$ with vanishing mean curvature on the boundary $\partial \varOmega $ must already be minimal graphs, which in particular yields some Bernstein-type result for Willmore graphs on $\mathbb{R }^2$ . Our methods also prove the non-existence of Willmore graphs on bounded $C^4$ -domains $\overline{\varOmega }$ with mean curvature $H$ satisfying $H \ge c_0>0 \,{\text{ on }}\, \partial \varOmega $ if $\varOmega $ contains some closed disc of radius $\frac{1}{c_0} \in (0,\infty )$ , and they yield that any closed Willmore surface in $\mathbb{R }^3$ which can be represented as a smooth graph over $\mathbb{S }^2$ has to be a round sphere. Finally, we demonstrate that our results are sharp by means of an examination of some certain part of the Clifford torus in $\mathbb{R }^3$ .     

On the third gap for proper holomorphic maps between balls

Let $F$ be a proper rational map from the complex ball $\mathbb B ^n$ into $\mathbb B ^N$ with $n>7$ and $3n+1 \le N\le 4n-7$ . Then $F$ is equivalent to a map $(G, 0, \dots , 0)$ where $G$ is a proper holomorphic map from $\mathbb B ^n$ into $\mathbb B ^{3n}$ .     

Symbolic extensions and dominated splittings for generic C^{1}-diffeomorphisms

Let $\mathrm{Diff }^1(M)$ be the set of all $C^1$ -diffeomorphisms $f:M\rightarrow M$ , where $M$ is a compact boundaryless d-dimensional manifold, $d\ge 2$ . We prove that there is a residual subset $\mathfrak R $ of $\mathrm{Diff }^1(M)$ such that if $f\in \mathfrak R $ and if $H(p)$ is the homoclinic class associated with a hyperbolic periodic point $p$ , then either $H(p)$ admits a dominated splitting of the form $E\oplus F_1\oplus \dots \oplus F_k\oplus G$ , where $F_i$ is not hyperbolic and one-dimensional, or $f|_{H(p)}$ has no symbolic extensions.     

Hyper-(rank one) groups

A group $G$ is called a $\mathcal{P }_1$ -group if it has a normal series of finite length whose factors have rank $1$ , while $G$ is an $\mathcal{H }_1$ -group if it has an ascending normal series of the same type. This paper investigates properties of $\mathcal{P }_1$ -groups and $\mathcal{H }_1$ -groups which correspond to known properties of nilpotent and supersoluble groups.     

On the Hall algebra of semigroup representations over \mathbb F _1

Let $\mathrm{A }$ be a finitely generated semigroup with 0. An $\mathrm{A }$ -module over $\mathbb F _1$ (also called an $\mathrm{A }$ -set), is a pointed set $(M,*)$ together with an action of $\mathrm{A }$ . We define and study the Hall algebra $\mathbb H _{\mathrm{A }}$ of the category $\mathcal C _{\mathrm{A }}$ of finite $\mathrm{A }$ -modules. $\mathbb H _{\mathrm{A }}$ is shown to be the universal enveloping algebra of a Lie algebra $\mathfrak n _{\mathrm{A }}$ , called the Hall Lie algebra of $\mathcal C _{\mathrm{A }}$ . In the case of $\langle t \rangle $ —the free monoid on one generator $\langle t \rangle $ , the Hall algebra (or more precisely the Hall algebra of the subcategory of nilpotent $\langle t \rangle $ -modules) is isomorphic to Kreimer’s Hopf algebra of rooted forests. This perspective allows us to define two new commutative operations on rooted forests. We also consider the examples when $\mathrm{A }$ is a quotient of $\langle t \rangle $ by a congruence, and the monoid $G \cup \{ 0\}$ for a finite group $G$ .     

Pfaffian representations of cubic surfaces

Let $\mathbb{K }$ be a field of characteristic zero. We describe an algorithm which requires a homogeneous polynomial $F$ of degree three in $\mathbb{K }[x_{0},x_1,x_{2},x_{3}]$ and a zero ${\mathbf{a }}$ of $F$ in $\mathbb{P }^{3}_{\mathbb{K }}$ and ensures a linear Pfaffian representation of $\text{ V}(F)$ with entries in $\mathbb{K }[x_{0},x_{1},x_{2},x_{3}]$ , under mild assumptions on $F$ and ${\mathbf{a }}$ . We use this result to give an explicit construction of (and to prove the existence of) a linear Pfaffian representation of $\text{ V}(F)$ , with entries in $\mathbb{K }^{\prime }[x_{0},x_{1},x_{2},x_{3}]$ , being $\mathbb{K }^{\prime }$ an algebraic extension of $\mathbb{K }$ of degree at most six. An explicit example of such a construction is given.     

Planar Beurling Transform and Bergman Type Spaces

In this paper we describe the actions of the operator $S_\mathbb{D }$ or its adjoint $S_\mathbb{D }^*$ on the poly-Bergman spaces of the unit disk $\mathbb{D }.$ Let $k$ and $j$ be positive integers. We prove that $(S_\mathbb{D })^{j}$ is an isometric isomorphism between the true poly-Bergman subspace $\mathcal{A }_{(k)}^2(\mathbb{D })\ominus N_{(k),j}$ onto the true poly-Bergman space $\mathcal{A }_{(j+k)}^2(\mathbb{D }),$ where the linear space $N_{(k),j}$ have finite dimension $j.$ The action of $(S_\mathbb{D })^{j-1}$ on the canonical Hilbert base for the Bergman subspace $\mathcal{A }^2(\mathbb{D })\ominus \mathcal{P }_{j-1},$ gives a Hilbert base $\{ \phi _{ j , k } \}_{ k }$ for $\mathcal{A }_{(j)}^2(\mathbb{D }).$ It is shown that $\{ \phi _{ j , k } \}_{ j, k }$ is a Hilbert base for $L^2(\mathbb{D },d A)$ such that whenever $j$ and $k$ remain constant we obtain a Hilbert base for the true poly-Bergman space $\mathcal{A }_{(j)}^2(\mathbb{D })$ and $\mathcal{A }_{(-k)}^2(\mathbb{D }),$ respectively. The functions $\phi _{ j , k }$ are polynomials in $z$ and $\overline{z}$ and are explicitly given in terms of the $(2,1)$ -hypergeometric polynomials. We prove explicit representations for the true poly-Bergman kernels and the Koshelev representation for the poly-Bergman kernels of $\mathbb{D }.$ The action of $S_\Pi $ on the true poly-Bergman spaces of the upper half-plane $\Pi $ allows one to introduce Hilbert bases for the true poly-Bergman spaces, and to give explicit representations of the true poly-Bergman and poly-Bergman kernels.     

Generalized normal homogeneous Riemannian metrics on spheres and projective spaces

In this paper, we develop new methods to study generalized normal homogeneous Riemannian manifolds. In particular, we obtain a complete classification of generalized normal homogeneous Riemannian metrics on spheres ${S^n}$ . We prove that for any connected (almost effective) transitive on $S^n$ compact Lie group $G$ , the family of $G$ -invariant Riemannian metrics on $S^n$ contains generalized normal homogeneous but not normal homogeneous metrics if and only if this family depends on more than one parameters and $n\ge 5$ . Any such family (that exists only for $n=2k+1$ ) contains a metric $g_\mathrm{can}$ of constant sectional curvature $1$ on $S^n$ . We also prove that $(S^{2k+1}, g_\mathrm{can})$ is Clifford–Wolf homogeneous, and therefore generalized normal homogeneous, with respect to $G$ (except the groups $G={ SU}(k+1)$ with odd $k+1$ ). The space of unit Killing vector fields on $(S^{2k+1}, g_\mathrm{can})$ from Lie algebra $\mathfrak g $ of Lie group $G$ is described as some symmetric space (except the case $G=U(k+1)$ when one obtains the union of all complex Grassmannians in $\mathbb{C }^{k+1}$ ).     

On additive involutions and Hamel bases

We provide an example of a discontinuous involutory additive function ${a: \mathbb{R}\to \mathbb{R}}$ such that ${a(H) \setminus H \ne \emptyset}$ for every Hamel basis ${H \subset \mathbb{R}}$ and show that, in fact, the set of all such functions is dense in the topological vector space of all additive functions from ${\mathbb{R}}$ to ${\mathbb{R}}$ with the Tychonoff topology induced by ${\mathbb{R}^{\mathbb{R}}}$ .