Actions of arithmetic groups on homology spheres and acyclic homology manifolds

We establish lower bounds on the dimensions in which arithmetic groups with torsion can act on acyclic manifolds and homology spheres. The bounds rely on the existence of elementary $p$ -groups in the groups concerned. In some cases, including ${\mathrm{Sp}}(2n,\mathbb Z )$ , the bounds we obtain are sharp: if $X$ is a generalized $\mathbb Z /3$ -homology sphere of dimension less than $2n-1$ or a $\mathbb Z /3$ -acyclic $\mathbb Z /3$ -homology manifold of dimension less than $2n$ , and if $n\ge 3$ , then any action of ${\mathrm{Sp}}(2n,\mathbb Z )$ by homeomorphisms on $X$ is trivial; if $n=2$ , then every action of ${\mathrm{Sp}}(2n,\mathbb Z )$ on $X$ factors through the abelianization of ${\mathrm{Sp}}(4,\mathbb Z )$ , which is $\mathbb Z /2$ .     

On the Topology of Kac–Moody groups

We study the topology of spaces related to Kac–Moody groups. Given a Kac–Moody group over $\mathbb C $ , let $\text {K}$ denote the unitary form with maximal torus ${{\mathrm{T}}}$ having normalizer ${{\mathrm{N}}}({{\mathrm{T}}})$ . In this article we study the cohomology of the flag manifold $\text {K}/{{{\mathrm{T}}}}$ as a module over the Nil-Hecke algebra, as well as the (co)homology of $\text {K}$ as a Hopf algebra. In particular, if $\mathbb F $ has positive characteristic, we show that $\text {H}_*(\text {K},\mathbb F )$ is a finitely generated algebra, and that $\text {H}^*(\text {K},\mathbb F )$ is finitely generated only if $\text {K}$ is a compact Lie group . We also study the stable homotopy type of the classifying space $\text {BK}$ and show that it is a retract of the classifying space $\text {BN(T)}$ of ${{\mathrm{N}}}({{\mathrm{T}}})$ . We illustrate our results with the example of rank two Kac–Moody groups.     

A zoo of diffeomorphism groups on \mathbb{R }^{n}

We consider the groups ${\mathrm{Diff }}_\mathcal{B }(\mathbb{R }^n)$ , ${\mathrm{Diff }}_{H^\infty }(\mathbb{R }^n)$ , and ${\mathrm{Diff }}_{\mathcal{S }}(\mathbb{R }^n)$ of smooth diffeomorphisms on $\mathbb{R }^n$ which differ from the identity by a function which is in either $\mathcal{B }$ (bounded in all derivatives), $H^\infty = \bigcap _{k\ge 0}H^k$ , or $\mathcal{S }$ (rapidly decreasing). We show that all these groups are smooth regular Lie groups.     

Value-sets of polynomials at p-adic integers

In this paper we are concerned with the classification of the subsets A of ${\mathbb{Z}_p}$ which occur as images ${f(\mathbb{Z}_p^r)}$ of polynomial functions ${f:\mathbb{Z}_p^r\to \mathbb{Z}_p}$ , limiting ourselves to compact-open subsets (i.e. finite unions of open balls). We shall prove three main results: (i) Every compact-open ${A\subset \mathbb{Z}_p}$ is of the shape ${A=f(\mathbb{Z}_p^r)}$ for suitable r and ${f\in\mathbb{Z}_p[X_1,\ldots ,X_r]}$ . (ii) For each r ₀ there is a compact-open A such that in (i) we cannot take r < r ₀. (iii) For any compact-open set ${A\subset \mathbb{Z}_p}$ there exists a polynomial ${f\in\mathbb{Q}_p[X]}$ such that ${f(\mathbb{Z}_p)=A}$ . We shall also discuss in more detail which sets A can be represented as ${f(\mathbb{Z}_p)}$ for a polynomial ${f\in\mathbb{Z}_p[X]}$ in a single variable.     

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 $G$ .     

2-Nilpotent real section conjecture

We show a $2$ -nilpotent section conjecture over $\mathbb{R }$ : for a geometrically connected curve $X$ over $\mathbb{R }$ such that each irreducible component of its normalization has $\mathbb{R }$ -points, $\pi _0(X(\mathbb{R }))$ is determined by the maximal $2$ -nilpotent quotient of the fundamental group with its Galois action, as the kernel of an obstruction of Jordan Ellenberg. This implies that for $X$ smooth and proper, $X(\mathbb{R })^{\pm }$ is determined by the maximal $2$ -nilpotent quotient of $\mathrm{Gal }(\mathbb{C }(X))$ with its $\mathrm{Gal }(\mathbb{R })$ action, where $X(\mathbb{R })^{\pm }$ denotes the set of real points equipped with a real tangent direction, showing a $2$ -nilpotent birational real section conjecture.     

Quadratic polynomials represented by norm forms

Let ${P(t) \in \mathbb{Q}[t]}$ be an irreducible quadratic polynomial and suppose that K is a quartic extension of ${\mathbb{Q}}$ containing the roots of P(t). Let ${{\bf N}_{K/\mathbb{Q}}({\rm x})}$ be a full norm form for the extension ${K/\mathbb{Q}}$ . We show that the variety $$\begin{array}{ll}P(t)={\bf N}_{K/\mathbb{Q}}({\rm x})\neq 0\end{array}$$ satisfies the Hasse principle and weak approximation. The proof uses analytic methods.     

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.     

Postulation of Disjoint Unions of Lines and Double Lines in {\mathbb{P}^3}

A double line ${C \subset \mathbb{P}^3}$ is a connected divisor of type (2, 0) on a smooth quadric surface. Fix ${(a, c) \in \mathbb{N}^2\ \backslash\ \{(0, 0)\}}$ . Let ${X \subset \mathbb{P}^3}$ be a general disjoint union of a lines and c double lines. Then X has maximal rank, i.e. for each ${t \in \mathbb{Z}}$ either ${h^1(\mathcal{I}_X(t)) = 0}$ or ${h^0(\mathcal{I}_X(t)) = 0}$ .     

p-Harmonic functions in the Heisenberg group: boundary behaviour in domains well-approximated by non-characteristic hyperplanes

In this paper we study, for given $p,~1<p<\infty $ , the boundary behaviour of non-negative $p$ -harmonic functions in the Heisenberg group $\mathbb{H }^n$ , i.e., we consider weak solutions to the non-linear and potentially degenerate partial differential equation $$\begin{aligned} \sum _{i=1}^{2n}X_i(|Xu|^{p-2}\,X_i u)=0 \end{aligned}$$ where the vector fields $X_1,\ldots ,X_{2n}$ form a basis for the space of left-invariant vector fields on $\mathbb{H }^n$ . In particular, we introduce a set of domains $\Omega \subset \mathbb{H }^n$ which we refer to as domains well-approximated by non-characteristic hyperplanes and in $\Omega $ we prove, for $2\le p<\infty $ , the boundary Harnack inequality as well as the Hölder continuity for ratios of positive $p$ -harmonic functions vanishing on a portion of $\partial \Omega $ .     

Global injectivity conditions for planar maps

We prove that a planar $C^1$ -smooth map $f:D\longrightarrow \mathbb{R }^{2n}$ , where $D\subseteq \mathbb{R }^{2n}$ is a convex open set, is injective if $\mathbb{R }\cap \mathrm{Spec}(df)_z=\emptyset $ for all $z\in D$ . We continue by showing that the triangulability of the differentials $(df)_z$ , $z\in D$ , ensure the global injectivity as well.     

Homogeneous almost quaternion-Hermitian manifolds

An almost quaternion-Hermitian structure on a Riemannian manifold $(M^{4n},g)$ is a reduction of the structure group of $M$ to $\mathrm{Sp}(n)\mathrm{Sp}(1)\subset \text{ SO }(4n)$ . In this paper we show that a compact simply connected homogeneous almost quaternion-Hermitian manifold of non-vanishing Euler characteristic is either a Wolf space, or $\mathbb{S }^2\times \mathbb{S }^2$ , or the complex quadric $\text{ SO }(7)/\mathrm{U}(3)$ .     

On Discrete groups of automorphisms of \mathbb P ^2_\mathbb{C }

We study the geometry and dynamics of discrete subgroups $\Gamma $ of $\mathrm{PSL}(3,\mathbb C )$ with an open invariant set $\Omega \subset \mathbb P _\mathbb{C }^2$ where the action is properly discontinuous and the quotient $\Omega /\Gamma $ contains a connected component whicis compact. We call such groups quasi-cocompact. In this case $\Omega /\Gamma $ is a compact complex projective orbifold and $\Omega $ is a divisible set. Our first theorem refines classical work by Kobayashi–Ochiai and others about complex surfaces with a projective structure: We prove that every such group is either virtually affine or complex hyperbolic. We then classify the divisible sets that appear in this way, the corresponding quasi-cocompact groups and the orbifolds $\Omega /\Gamma $ . We also prove that excluding a few exceptional cases, the Kulkarni region of discontinuity coincides with the equicontinuity region and is the largest open invariant set where the action is properly discontinuous.     

Hurwitz-type bound, knot surgery, and smooth {\mathbb{S }}^1-four-manifolds

In this paper we prove several related results concerning smooth $\mathbb{Z }_p$ or $\mathbb{S }^1$ actions on $4$ -manifolds. We show that there exists an infinite sequence of smooth $4$ -manifolds $X_n$ , $n\ge 2$ , which have the same integral homology and intersection form and the same Seiberg-Witten invariant, such that each $X_n$ supports no smooth $\mathbb{S }^1$ -actions but admits a smooth $\mathbb{Z }_n$ -action. In order to construct such manifolds, we devise a method for annihilating smooth $\mathbb{S }^1$ -actions on $4$ -manifolds using Fintushel-Stern knot surgery, and apply it to the Kodaira-Thurston manifold in an equivariant setting. Finally, the method for annihilating smooth $\mathbb{S }^1$ -actions relies on a new obstruction we derived in this paper for existence of smooth $\mathbb{S }^1$ -actions on a $4$ -manifold: the fundamental group of a smooth $\mathbb{S }^1$ -four-manifold with nonzero Seiberg-Witten invariant must have infinite center. We also include a discussion on various analogous or related results in the literature, including locally linear actions or smooth actions in dimensions other than four.     

Computing α-Invariants of Singular del Pezzo Surfaces

We prove a new local inequality for divisors on surfaces and utilize it to compute α-invariants of singular del Pezzo surfaces, which implies that del Pezzo surfaces of degree one whose singular points are of type $\mathbb{A}_{1}$ , $\mathbb{A}_{2}$ , $\mathbb{A}_{3}$ , $\mathbb{A}_{4}$ , $\mathbb{A}_{5}$ , or $\mathbb{A}_{6}$ are Kähler-Einstein.     

Categories of bounded \left( {\mathfrak{sp}\left( {{{\mathrm{S}}^2}V \oplus {{\mathrm{S}}^2}{V^{*}}} \right),\;\mathfrak{gl}(V)} \right) - and \left( {\mathfrak{sp}\left( {{\varLambda^2}V \oplus {\varLambda^2}{V^{*}}} \right),\;\mathfrak{gl}(V)} \right) -modules

Let $ \mathfrak{g} $ be a reductive Lie algebra over $ \mathbb{C} $ and $ \mathfrak{k} \subset \mathfrak{g} $ be a reductive in $ \mathfrak{g} $ subalgebra. We call a $ \mathfrak{g} $ -module M a $ \left( {\mathfrak{g}{\hbox{,}}\;\mathfrak{k}} \right) $ -module whenever M is a direct sum of finite-dimensional $ \mathfrak{k} $ -modules. We call a $ \left( {\mathfrak{g}{\hbox{,}}\;\mathfrak{k}} \right) $ -module M bounded if there exists $ {C_M} \in {\mathbb{Z}_{{ \geqslant 0}}} $ such that for any simple finite-dimensional $ \mathfrak{k} $ -module E the dimension of the E-isotypic component is not greater than C _M dim E. Bounded $ \left( {\mathfrak{g}{\hbox{,}}\;\mathfrak{k}} \right) $ -modules form a subcategory of the category of $ \mathfrak{g} $ -modules. Let V be a finite-dimensional vector space. We prove that the categories of bounded $ \left( {\mathfrak{sp}\left( {{{\mathrm{S}}^2}V \oplus {{\mathrm{S}}^2}{V^{*}}} \right),\;\mathfrak{gl}(V)} \right) $ - and $ \left( {\mathfrak{sp}\left( {{\varLambda^2}V \oplus {\varLambda^2}{V^{*}}} \right),\;\mathfrak{gl}(V)} \right) $ -modules are isomorphic to the direct sum of countably many copies of the category of representations of some explicitly described quiver with relations under some mild assumptions on the dimension of V .     

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.     

Cubic symmetric polynomials yielding variations of the Erdős–Ginzburg–Ziv theorem

Let p be a prime and let $\varphi\in\mathbb{Z}_{p}[x_{1},x_{2},\ldots, x_{p}]$ be a symmetric polynomial, where  $\mathbb {Z}_{p}$ is the field of p elements. A sequence T in  $\mathbb {Z}_{p}$ of length p is called a φ-zero sequence if φ(T)=0; a sequence in $\mathbb {Z}_{p}$ is called a φ-zero free sequence if it does not contain any φ-zero subsequence. Motivated by the EGZ theorem for the prime p, we consider symmetric polynomials $\varphi\in \mathbb {Z}_{p}[x_{1},x_{2},\ldots, x_{p}]$ , which satisfy the following two conditions: (i) every sequence in  $\mathbb {Z}_{p}$ of length 2p?1 contains a φ-zero subsequence, and (ii) the φ-zero free sequences in  $\mathbb {Z}_{p}$ of maximal length are all those containing exactly two distinct elements, where each element appears p?1 times. In this paper, we determine all symmetric polynomials in $\mathbb {Z}_{p}[x_{1},x_{2},\ldots, x_{p}]$ of degree not exceeding 3 satisfying the conditions above.     

Fundamental Solutions in the Colombeau Framework: Applications to Solvability and Regularity Theory

In this article we introduce the notion of fundamental solution in the Colombeau context as an element of the dual $\mathcal {L}(\ensuremath {\mathcal {G}_{\mathrm{c}}}(\mathbb {R}^{n}),\widetilde {\mathbb {C}})$ . After having proved the existence of a fundamental solution for a large class of partial differential operators with constant Colombeau coefficients, we investigate the relationships between fundamental solutions in $\mathcal {L}(\ensuremath {\mathcal {G}_{\mathrm{c}}}(\mathbb {R}^{n}),\widetilde {\mathbb {C}})$ , Colombeau solvability and $\ensuremath {\mathcal {G}}$ - and $\ensuremath {\ensuremath {\mathcal {G}}^{\infty}}$ -hypoellipticity respectively.