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The modality of a Borel subgroup in a simple algebraic group of type $E_8$Feb 09 2018Let $G$ be a simple algebraic group over an algebraically closed field $k$, where $\mathrm{char}\, k$ is either 0 or a good prime for $G$. We consider the modality $\mathrm{mod}(B : \mathfrak u)$ of the action of a Borel subgroup $B$ of $G$ on the Lie ... More

Graphes associés au groupe de CremonaFeb 08 2018To reinforce the analogy between the mapping class group and the Cremona group of rank 2 over an algebraic closed field, we look for a graph analoguous to the curve graph and such that the Cremona group acts on it non-trivially. The first candidate is ... More

Positivity and higher Teichmüller theoryFeb 08 2018We introduce $\Theta$-positivity, a new notion of positivity in real semisimple Lie groups. The notion of $\Theta$-positivity generalizes at the same time Lusztig's total positivity in split real Lie groups as well as well known concepts of positivity ... More

Gromov-Hyperbolicity of the ray graph and quasimorphisms on a big mapping class groupFeb 08 2018These notes are the English version of the paper "Hyperbolicit\'e du graphe des rayons et quasi-morphismes sur un gros groupe modulaire". The mapping class group Gamma of the complement of a Cantor set in the plane arises naturally in dynamics. We show ... More

Normal elements of completed group algebras over ${\rm SL}_3(\mathbb{Z}_p) $Feb 08 2018Let $p$ be a prime integer and $\mathbb{Z}_p$ be the ring of $p$-adic integers. By a purely computational approach we prove that each nonzero normal element of a completed group algebra over the special linear group ${\rm SL}_3(\mathbb{Z}_p)$ is a unit. ... More

Samuel compactifications of automorphism groupsFeb 07 2018We study the Samuel compactification of the automorphism group of a countable first-order structure. A key motivating question is a problem of Ellis and a counter-conjecture by Pestov regarding the difference between $S(G)$, the Samuel compactification, ... More

The intrinsic geometry of coarse median spaces and their intervalsFeb 07 2018Sholander showed that median algebras can be characterised in terms of the intervals arising from the median operator. In this paper we develop a coarse analogue of Sholander's approach, characterising coarse median spaces in terms of their intervals. ... More

Asymptotically Locally Euclidean/Kaluza-Klein Stationary Vacuum Black Holes in 5 DimensionsFeb 07 2018We produce new examples, both explicit and analytical, of bi-axisymmetric stationary vacuum black holes in 5 dimensions. A novel feature of these solutions is that they are asymptotically locally Euclidean in which spatial cross-sections at infinity have ... More

Canonicality of Makanin-Razborov Diagrams - CounterexampleFeb 07 2018Sets of solutions to finite systems of equations in a free group, are equivalent to sets of homomorphisms from a fixed f.p. group into a free group. The latter can be encoded in a diagram, the construction of which is valid also for f.g. groups. The diagram ... More

A group-theoretical interpretation of the word problem for free idempotent generated semigroupsFeb 07 2018The set of idempotents of any semigroup carries the structure of a biordered set, which contains a great deal of information concerning the idempotent generated subsemigroup of the semigroup in question. This leads to the construction of a free idempotent ... More

Semi Concurrent vector fields in Finsler geometryFeb 07 2018In the present paper, we introduce and investigate the notion of a semi concurrent vector field on a Finsler manifold. We show that some special Finsler manifolds admitting such vector fields turn out to be Riemannian. We prove that Tachibana's characterization ... More

Stratifications of affine Deligne-Lusztig varietiesFeb 06 2018Affine Deligne-Lusztig varieties are analogues of Deligne-Lusztig varieties in the context of affine flag varieties and affine Grassmannians. They are closely related to moduli spaces of $p$-divisible groups in positive characteristic, and thus to arithmetic ... More

On the length and depth of finite groups (with an appendix by D.R. Heath-Brown)Feb 06 2018An unrefinable chain of a finite group $G$ is a chain of subgroups $G = G_0 > G_1 > \cdots > G_t = 1$, where each $G_i$ is a maximal subgroup of $G_{i-1}$. The length (respectively, depth) of $G$ is the maximal (respectively, minimal) length of such a ... More

Crossed extensions and equivalences of topological 2-groupoidsFeb 06 2018We provide concrete models for generalized morphisms and Morita equivalences of topological 2-groupoids by introducing the notions of crossings and crossed extensions of groupoid crossed modules. A systematic study of these objects is elaborated and an ... More

On the structure of random graphs that are locally indistinguishable from a latticeFeb 06 2018We study the properties of finite graphs in which the ball of radius $r$ around each vertex induces a graph isomorphic to the ball of radius $r$ in some fixed vertex-transitive graph $F$. This is a natural extension of the study of regular graphs. We ... More

Multiplicativity of the idempotent splittings of the Burnside ring and the G-sphere spectrumFeb 06 2018We provide a complete characterization of the equivariant commutative ring structures of all the factors in the idempotent splitting of the G-equivariant sphere spectrum, including their Hill-Hopkins-Ravenel norms, where G is any finite group. Our results ... More

On the width of transitive sets: bounds on matrix coefficients of finite groupsFeb 06 2018We say that a finite subset of the unit sphere in $\mathbf{R}^d$ is transitive if there is a group of isometries which acts transitively on it. We show that the width of any transitive set is bounded above by a constant times $(\log d)^{-1/2}$. This is ... More

A new construction of CAT(0) cube complexesFeb 06 2018We introduce the notion of cube complex with coupled link (CLCC) as a mean of constructing interesting CAT(0) cubulated groups. CLCCs are defined locally, making them a useful tool to use when precise control over the links is required. In this paper ... More

Conformal Parametrisation of Loxodromes by Triples of CirclesFeb 06 2018We provide a parametrisation of a loxodrome by three specially arranged cycles. The parametrisation is covariant under fractional linear transformations of the complex plane and naturally encodes conformal properties of loxodromes. Selected geometrical ... More

Renormalization for a Scalar Field in an External Scalar PotentialFeb 05 2018The Pauli--Villars regularization procedure confirms and sharpens the conclusions reached previously by covariant point splitting. The divergences in the stress tensor of a quantized scalar field interacting with a static scalar potential are isolated ... More

Commutator Subgroups of Virtual and Welded Braid GroupsFeb 05 2018Let $VB_n$, resp. $WB_n$ denote the virtual, resp. welded, braid group on $n$ strands. We study their commutator subgroups $VB_n' = [VB_n, VB_n]$ and, $WB_n' = [WB_n, WB_n]$ respectively. We obtain a set of generators and defining relations for these ... More

Unitarity issues in higher derivative field theoriesFeb 03 2018We analyze the unitarity properties of higher derivative quantum field theories which are free of ghosts and ultraviolet singularities. We point out that in spite of the absence of ghosts most of these theories are not unitary. This result confirms the ... More

Brauer characters and normal Sylow $p$-subgroupsFeb 03 2018In this paper, we study some variations of the well-known It\^{o}-Michler theorem for $p$-Brauer characters using various inequalities involving the $p$-Brauer character degrees of finite groups. Several new criteria for the existence of a normal Sylow ... More

Small sphere limit of the quasi-local energy with anti de-Sitter space referenceFeb 02 2018In [13], a new quasi-local energy is introduced for spacetimes with a non-zero cosmological constant. In this article, we study the small sphere limit of this newly defined quasi-local energy for spacetimes with a negative cosmological constant. For such ... More

Mode solutions for a Klein-Gordon field in anti-de Sitter with dynamical boundary conditions of Wentzell typeFeb 01 2018We study a real, massive Klein-Gordon field in the Poincar\'e fundamental domain of the $(d+1)$-dimensional anti-de Sitter (AdS) spacetime, subject to a particular choice of dynamical boundary conditions of generalized Wentzell type, whereby the boundary ... More

The Multiple Holomorphs of Finite $p$-Groups of Class TwoJan 31 2018$\DeclareMathOperator{\Hol}{Hol}$$\DeclareMathOperator{\Aut}{Aut}$$\newcommand{\Gp}[0]{\mathcal{G}(p)}$Let $G$ be a group, and $S(G)$ be the group of permutations on the set $G$. Define the holomorph of $G$ to be the normalizer of the image in $S(G)$ ... More

Invariable generation of permutation and linear groupsJan 30 2018A subset $\left\{x_{1},x_{2},\hdots,x_{d}\right\}$ of a group $G$ \emph{invariably generates} $G$ if $\left\{x_{1}^{g_{1}},x_{2}^{g_{2}},\hdots,x_{d}^{g_{d}}\right\}$ generates $G$ for every $d$-tuple $(g_{1},g_{2}\hdots,g_{d})\in G^{d}$. We prove that ... More

An Algorithm to Decompose Permutation Representations of Finite Groups: Polynomial Algebra ApproachJan 29 2018We describe an algorithm for splitting a permutation representation of a finite group into irreducible components. The algorithm is based on the fact that the components of the invariant inner product in invariant subspaces are operators of projection ... More

The number of homomorphisms from the Hawaiian earring groupJan 29 2018We show a dichotomy for groups of cardinality less than continuum. The number of homomorphisms from the Hawaiian earring group to such a group $G$ is either the cardinality of $G$ in case $G$ is noncommutatively slender, or the number is $2^{2^{\aleph_0}}$ ... More

Hyperbolicity and Cubulability Are Preserved Under Elementary EquivalenceJan 29 2018The following properties are preserved under elementary equivalence, among finitely generated groups: being hyperbolic (possibly with torsion), being hyperbolic and cubulable, and being a subgroup of a hyperbolic group. In other words, if a finitely generated ... More

Operational calculus for Fourier transform on the group $GL(2,R)$Jan 29 2018Consider the Fourier transform on the group $GL(2,R)$ of real $2\times 2$-matrices. We show that Fourier-images of polynomial differential operators on $GL(2,R)$ are differential-difference operators with coefficients meromorphic in parameters of representations. ... More

Reducible subgroups of exceptional algebraic groupsJan 28 2018Let $G$ be a simple algebraic group over an algebraically closed field. A closed subgroup $H$ of $G$ is called $G$-completely reducible ($G$-cr) if, whenever $H$ is contained in a parabolic subgroup $P$ of $G$, it is contained in a Levi factor of $P$. ... More

Finite Groups Havinfg Nonnormal T.I. SubgroupsJan 28 2018In the present paper, the structure of a finite group $G$ having a nonnormal T.I. subgroup $H$ which is also a Hall $\pi$-subgroup is studied. As a generalization of a result due to Gow, we prove that $H$ is a Frobenius complement whenever $G$ is $\pi$-separable. ... More

On Uniform Admissibility of Unitary and Smooth RepresentationsJan 26 2018Let $G$ be a locally compact totally disconnected topological group. Under a necessary mild assumption, we show that the irreducible unitary representations of $G$ are uniformly admissible if and only if the irreducible smooth representations of $G$ are ... More

Stability and Invariant Random SubgroupsJan 25 2018Consider $\operatorname{Sym}(n)$, endowed with the normalized Hamming metric $d_n$. A finitely-generated group $\Gamma$ is \emph{P-stable} if every almost homomorphism $\rho_{n_k}\colon \Gamma\rightarrow\operatorname{Sym}(n_k)$ (i.e., for every $g,h\in\Gamma$, ... More

Maximal subgroups of ${}^2E_6(2)$ and its automorphism groupsJan 25 2018We give a new computer-assisted proof of the classification of maximal subgroups of the simple group ${}^2E_6(2)$ and its extensions by any subgroup of the outer automorphism group $S_3$. This is not a new result, but no earlier proof exists in the literature. ... More

On Behaviors of Maximal DimensionJan 25 2018Jan 26 2018In this paper, we investigate behaviors of Maximal Dimension, a group invariant involving certain configuration of maximal subgroups, which we denote by MaxDim. We prove that in some special cases, MaxDim(G\times H) = MaxDim(G) + MaxDim(H). We also prove ... More

Energy-parity from a bicomplex algebraJan 24 2018By replacing the field of complex numbers with the commutative ring of bicomplex numbers, we attempt to construct interacting scalar quantum field theories that feature both positive- and negative-energy states. This work places the tentative ideas proposed ... More

Overgroups of exterior powers of an elementary group. I. Levels and normalizersJan 24 2018In the present paper, we prove the first part in the standard description of groups $H$ lying between $m$-th exterior power of elementary group $E(n,R)$ and the general linear group $GL_{\binom{n}{m}}(R)$. We study structure of the exterior power of elementary ... More

Axisymmetric black holes allowing for separation of variables in the Klein-Gordon and Hamilton-Jacobi equationJan 22 2018Jan 23 2018We determine the class of axisymmetric and asymptotically flat black-hole spacetimes for which the test Klein-Gordon and Hamilton-Jacobi equations allow for the separation of variables. The known Kerr, Kerr-Newman, Kerr-Sen and some other black-hole metrics ... More

Counting subrings of the ring $\Bbb{Z}_m \times \Bbb{Z}_n$Jan 22 2018Let $m,n\in \Bbb{N}$. We represent the additive subgroups of the ring $\Bbb{Z}_m \times \Bbb{Z}_n$, which are also (unital) subrings, and deduce explicit formulas for $N^{(s)}(m,n)$ and $N^{(us)}(m,n)$, denoting the number of subrings of the ring $\Bbb{Z}_m ... More

Generically free representations III: exceptionally bad characteristicJan 22 2018In parts I and II, we determined which irreducible representations $V$ of a simple linear algebraic group $G$ are generically free for Lie($G$), i.e., which $V$ have an open subset consisting of vectors whose stabilizer in Lie($G$) is zero, with some ... More

Global and Concrete Quantizations on General Type I GroupsJan 21 2018In recent papers and books, a global quantization has been developed for {\it unimodular} groups of type I\,. It involves operator-valued symbols defined on the product between the group $\G$ and its unitary dual $\wG$\,, composed of equivalence classes ... More

On the asymptotic behavior of static perfect fluidsJan 20 2018Static spherically symmetric solutions to the Einstein-Euler equations with prescribed central densities are known to exist, be unique and smooth for reasonable equations of state. Some criteria are also available to decide whether solutions have finite ... More

Scalar-torsion theories of gravity I: general formalism and conformal transformationsJan 19 2018We discuss the most general class of teleparallel scalar-torsion theories of gravity in their covariant formulation. The only restrictions we impose are the invariance of the action under diffeomorphisms and local Lorentz transformations, as well as vanishing ... More

Groups whose elements are not conjugate to their powersJan 18 2018We call a finite group irrational if none of its elements is conjugate to a distinct power of itself. We prove that those groups are solvable and describe certain classes of these groups, where the above property is only required for $p$-elements, for ... More

The shape of solvable groups with odd orderJan 17 2018We prove that the minimal composition length, $c$, of a solvable group with solvable length $d$ satisfies $9^{(d-3)/9}< c< 9^{(d+1)/5}$, and the minimal composition length, $c^o$, of a group with odd order and solvable length $d$ satisfies $7^{(d-2)/5}< ... More

Presentations of parabolics in some elementary Chevalley-Demazure groupsJan 16 2018Given a universal elementary Chevalley-Demazure group $E_\Phi^{sc}(R)$ for which its (standard) parabolic subgroups are finitely generated, we consider the problem of classifying which parabolics $P(R) \subset E_\Phi^{sc}(R)$ are finitely presented. We ... More

Normal subgroups of the braid group and the metaconjecture of IvanovJan 16 2018We show that many normal subgroups of the braid group modulo its centre, and of the mapping class group of a sphere with marked points, have the property that their automorphism and abstract commensurator groups are mapping class groups of such spheres. ... More

Modular group algebras whose group of unitary units is locally nilpotentJan 16 2018We characterize those modular group algebras FG whose group of unitary units is locally nilpotent under the classical involution of FG.

Group algebra whose unit group is locally nilpotentJan 16 2018We present a complete list of groups $G$ and fields $F$ for which: (i) the group of normalized units V(FG) of the group algebra FG is locally nilpotent; (ii) the group algebra FG has a finite number of nilpotent elements and V(FG) is an Engel group.

Pointlike sets for varieties determined by groupsJan 15 2018For a variety of finite groups $\mathbf H$, let $\overline{\mathbf H}$ denote the variety of finite semigroups all of whose subgroups lie in $\mathbf H$. We give a characterization of the subsets of a finite semigroup that are pointlike with respect to ... More

Regular orbits of sporadic simple groupsJan 14 2018Given a finite group $G$ and a faithful irreducible $FG$-module $V$ where $F$ has prime order, does $G$ have a regular orbit on $V$? This problem is equivalent to determining which primitive permutation groups of affine type have a base of size 2. We ... More

On Congruence Permutable $G$-setsJan 14 2018An algebraic structure is said to be congruence permutable if its arbitrary congruences $\alpha$ and $\beta$ satisfy the equation $\alpha \circ \beta =\beta \circ \alpha$, where $\circ$ denotes the usual composition of binary relations. For an arbitrary ... More

2D problems in groupsJan 13 2018Jan 22 2018We investigate a conjecture about stabilisation of deficiency in finite index subgroups and relate it to the D2 Problem of C.T.C. Wall and the Relation Gap problem. We verify the pro-$p$ version of the conjecture, as well as its higher dimensional abstract ... More

Homogeneous length functions on groupsJan 11 2018A pseudo-length function defined on an arbitrary group $G = (G,\cdot,e, (\,)^{-1})$ is a map $\ell : G \to [0,+\infty)$ obeying $\ell(e)=0$, the symmetry property $\ell(x^{-1}) = \ell(x)$, and the triangle inequality $\ell(xy) \leqslant \ell(x) + \ell(y)$ ... More

Smooth Version of Johnson's Problem Concerning Derivations of Group AlgebrasJan 10 2018A description of the algebra of outer derivations of a group algebra of a finitely presented discrete group is given in terms of the Cayley complex of the groupoid of the adjoint action of the group. This task is a smooth version of Johnson's problem ... More

Fixed points of automorphisms of certain non-cyclic $p$-groups and the dihedral groupJan 10 2018Jan 18 2018Let $G=\mathbf{Z}_{p} \oplus \mathbf{Z}_{p^2}$ where $p$ is a prime number. Suppose that $d$ is a divisor of $G$. In this paper we find the number of automorphisms of $G$ fixing $d$ elements of $G$, and denote it by $\theta(G,d)$. As a consequence, we ... More

Closures of locally divergent orbits of maximal tori and values of homogeneous formsJan 05 2018Let $\G$ be a semisimple algebraic group over a number field $K$, $\mathcal{S}$ a finite set of places of $K$, $K_\mathcal{S}$ the direct product of the completions $K_v, v \in \mathcal{S}$, and $\OO$ the ring of $\mathcal{S}$-integers of $K$. Let $G ... More

New integrable models and analytical solutions in $f(R)$~cosmology with an ideal gasJan 04 2018In the context of $f\left( R\right) $-gravity with a spatially flat FLRW metric containing an ideal fluid, we use the method of invariant transformations to specify families of models which are integrable. We find three families of $f(R)$ theories for ... More

Conjectures on the Relations of Linking and Causality in Causally Simple SpacetimesDec 28 2017We formulate the generalization of the Legendrian Low conjecture of Natario and Tod (proved by Nemirovski and myself before) to the case of causally simple spacetimes. We prove a weakened version of the corresponding statement. In all known examples, ... More

Normalisers of parabolic subgroups in finite unitary reflection groupsDec 27 2017It is well known that the normaliser of a parabolic subgroup of a finite Coxeter group is the semidirect product of the parabolic subgroup by the stabiliser of a set of simple roots. We show that a similar result holds for all finite unitary reflection ... More

Bianchi cosmologies with $p$-form gauge fieldsDec 23 2017In this paper the dynamics of free gauge fields in Bianchi type I-VII$_{h}$ space-times is investigated. The general equations for a matter sector consisting of a $p$-form field strength ($p\,\in\,\{1,3\}$), a cosmological constant ($4$-form) and perfect ... More

Teleparallel theories of gravity as analogue of non-linear electrodynamicsNov 27 2017The teleparallel formulation of gravity theories reveals close structural analogies to electrodynamics, which are more hidden in their usual formulation in terms of the curvature of spacetime. We show how every locally Lorentz invariant teleparallel theory ... More

Lorentzian length spacesNov 24 2017We introduce an analogue of the theory of length spaces into the setting of Lorentzian geometry and causality theory. The r\^ole of the metric is taken over by the time separation function, in terms of which all basic notions are formulated. In this way ... More

Physical models from noncommutative causalityNov 14 2017We introduced few years ago a new notion of causality for noncommutative spacetimes directly related to the Dirac operator and the concept of Lorentzian spectral triple. In this paper, we review in a non-technical way the noncommutative causal structure ... More

The Lorentzian distance formula in noncommutative geometryOct 30 2017Jan 24 2018For almost twenty years, a search for a Lorentzian version of the well-known Connes' distance formula has been undertaken. Several authors have contributed to this search, providing important milestones, and the time has now come to put those elements ... More

A Counterexample to the First Zassenhaus ConjectureOct 24 2017Nov 20 2017Hans J. Zassenhaus conjectured that for any unit $u$ of finite order in the integral group ring of a finite group $G$ there exists a unit $a$ in the rational group algebra of $G$ such that $a^{-1}\cdot u \cdot a=\pm g$ for some $g\in G$. We disprove this ... More

Groups of order $p^9$, class 2, and exponent $p$ having derived group of order $p^2$Oct 15 2017This paper concerns finite groups of class (at most) two and of odd prime exponent $p$. Such a group is called special if the center lies within its derived group. Every group of class 2 and exponent $p$ can be uniquely expressed as the direct product ... More

On quasi-isometric nilpotent Lie groupsOct 12 2017Nov 20 2017In this article we provide evidence for a well-known conjecture which states that quasi-isometric simply-connected nilpotent Lie groups are isomorphic. We do so by constructing new examples which are rigid in the sense that whenever they are quasi-isometric ... More

Wave and Dirac equations on manifoldsOct 12 2017We review some recent results on geometric equations on Lorentzian manifolds such as the wave and Dirac equations. This includes well-posedness and stability for various initial value problems, as well as results on the structure of these equations on ... More

No iterated identities satisfied by all finite groupsOct 11 2017Dec 07 2017We show that there is no iterated identity satisfied by all finite groups. For $w$ being a non-trivial word of length $l$, we show that there exists a finite group $G$ of cardinality at most $\exp(l^C)$ which does not satisfy the iterated identity $w$. ... More

Stability of the Kasner Universe in f(T) GravitySep 11 2017Jan 12 2018$f(T)$ gravity offers an alternative context in which to consider gravitational interactions where torsion, rather than curvature, is the mechanism by which gravitation is communicated. We investigate the stability of the Kasner solution with several ... More

Automorphism groups of finite topological rankSep 06 2017We offer a criterion for showing that the automorphism group of an ultrahomogeneous structure is topologically 2-generated and even has a cyclically dense conjugacy class. We then show how finite topological rank of the automorphism group of an $\omega$-categorical ... More

Aproximación métrica de grupos: una breve perspectivaSep 05 2017Sep 06 2017This is an expository paper (in Spanish) about the metric approximation of groups.

Kitaev model for mirror bicrossproduct Hopf algebras and its tensor network representationSep 02 2017Kitaev's model [1,2] is usually defined in terms of the Drinfeld double. We propose a new version defined in terms of the mirror bicrossproduct quantum group [3]. By some aspects, even though the bicrossproduct quantum group is more complicated than the ... More

Classical r-matrices for the generalised Chern-Simons formulation of 3d gravityAug 25 2017We study the conditions for classical r-matrices to be compatible with the generalised Chern-Simons action for 3d gravity. Compatibility means solving the classical Yang-Baxter equations with a prescribed symmetric part for each of the Lie algebras and ... More

The $K(π, 1)$-problem for restrictions of complex reflection arrangementsAug 17 2017In 1962, Fadell and Neuwirth showed that the configuration space of the braid arrangement is aspherical. Having generalized this to many real reflection groups, Brieskorn conjectured this for all Coxeter groups. This follows from Deligne's seminal work ... More

Derivations of Group AlgebrasAug 16 2017In the paper, a method of describing the outer derivations of the group algebra of a finitely presentable group is given. The description of derivations is given in terms of characters of the groupoid of the adjoint action of the group.

Isoperimetric inequalities, shapes of Følner sets and groups with Shalom's property ${H_{\mathrm{FD}}}$Aug 16 2017Oct 19 2017We prove an isoperimetric inequality for groups. As an application, we obtain lower bound on F{\o}lner functions in various nilpotent-by-cyclic groups. Under a regularity assumption, we obtain a characterization of F{\o}lner functions of these groups. ... More

Subquandles of affine quandlesAug 08 2017A quandle will be called quasi-affine, if it embeds into an affine quandle. Our main result is a characterization of quasi-affine quandles, by group-theoretic properties of their displacement group, by a universal algebraic condition coming from the commutator ... More

Set-Direct Factorizations of GroupsJul 14 2017We consider factorizations $G=XY$ where $G$ is a general group, $X$ and $Y$ are normal subsets of $G$ and any $g\in G$ has a unique representation $g=xy$ with $x\in X$ and $y\in Y$. This definition coincides with the customary and extensively studied ... More

Warped Product Space-timesJul 05 2017Many classical results in relativity theory concerning spherically symmetric space-times have easy generalizations to warped product space-times, with a two-dimensional Lorentzian base and arbitrary dimensional Riemannian fibers. We first give a systematic ... More

Semisimple Weakly Symmetric Pseudo--Riemannian ManifoldsJul 04 2017Jan 10 2018We develop the classification of weakly symmetric pseudo--riemannian manifolds $G/H$ where $G$ is a semisimple Lie group and $H$ is a reductive subgroup. We derive the classification from the cases where $G$ is compact, and then we discuss the (isotropy) ... More

Cellular Automata on Group SetsJun 26 2017We introduce and study cellular automata whose cell spaces are left-homogeneous spaces. Examples of left-homogeneous spaces are spheres, Euclidean spaces, as well as hyperbolic spaces acted on by isometries; uniform tilings acted on by symmetries; vertex-transitive ... More

Finite groups with large Noether number are almost cyclicJun 26 2017Jul 13 2017Noether, Fleischmann and Fogarty proved that if the characteristic of the underlying field does not divide the order $|G|$ of a finite group $G$, then the polynomial invariants of $G$ are generated by polynomials of degrees at most $|G|$. Let $\beta(G)$ ... More

The Moore and the Myhill Property For Strongly Irreducible Subshifts Of Finite Type Over Group SetsJun 19 2017We prove the Moore and the Myhill property for strongly irreducible subshifts over right amenable and finitely right generated left homogeneous spaces with finite stabilisers. Both properties together mean that the global transition function of each big-cellular ... More

Ehlers-Kundt Conjecture about Gravitational Waves and Dynamical SystemsJun 12 2017Nov 01 2017Ehlers-Kundt conjecture is a physical assertion about the fundamental role of plane waves for the description of gravitational waves. Mathematically, it becomes equivalent to a problem on the Euclidean plane ${\mathbb R}^2$ with a very simple formulation ... More

Examples of finite free complexes of small rank and homologyJun 07 2017This work concerns finite free complexes over commutative noetherian rings, in particular over group algebras of elementary abelian groups. The main contribution is the construction of complexes such that the ranks of their underlying free modules, or ... More

Magnetic Zero-Modes, Vortices and Cartan GeometryMay 26 2017Nov 14 2017We show that magnetic zero-modes of the Dirac operator on $\mathbb{R}^3$ which obey an additional non-linear equation are closely related to vortex configurations on the 2-sphere, and that both are best understood in terms of the geometry induced on the ... More

The power graph of a torsion-free groupMay 03 2017Jan 29 2018The \emph{power graph} $P(G)$ of a group $G$ is the graph whose vertex set is $G$, with $x$ and $y$ joined if one is a power of the other; the \emph{directed power graph} $\vec{P}(G)$ has the same vertex set, with an arc from $x$ to $y$ if $y$ is a power ... More

The $L^2$-torsion polytope of amenable groupsApr 24 2017Jan 29 2018We introduce the notion of groups of polytope class and show that torsion-free amenable groups satisfying the Atiyah Conjecture possess this property. A direct consequence is the homotopy invariance of the $L^2$-torsion polytope among $G$-CW-complexes ... More

Constant scalar curvature hypersurfaces in $(3+1)$-dimensional GHMC Minkowski spacetimesMar 26 2017Nov 22 2017We prove that every $(3+1)$-dimensional flat GHMC Minkowski spacetime which is not a translation spacetime or a Misner spacetime carries a unique foliation by spacelike hypersurfaces of constant scalar curvature. In otherwords, we prove that every such ... More

On 3+1 Lorentzian Einstein Manifolds with One Rotational IsometryMar 18 2017We consider 3+1 rotationally symmetric Lorentzian Einstein spacetime manifolds with $\Lambda >0$ and reduce the equations to 2+1 Einstein equations coupled to `shifted' wave maps. Subsequently, we prove various (explicit) positive mass-energy theorems. ... More

A short note about diffuse Bieberbach groupsMar 15 2017Sep 26 2017We consider low dimensional diffuse Bieberbach groups. In particular we classify diffuse Bieberbach groups up to dimension 6. We also answer a question from [S. Kionke, J. Raimbault, On geometric aspects of diffuse groups, Doc. Math. 21 (2016), page 887] ... More

Normality in Hopf Galois extensionsMar 07 2017For a Hopf Galois structure on a finite separable field extension, we prove a result which generalizes the fact that intermediate fields of a Galois field extension corresponding to normal subgroups of the Galois group are Galois extensions of the base ... More

Criticality of inhomogeneous Nariai-like cosmological modelsFeb 09 2017Apr 25 2017In this paper, we construct and study solutions of Einstein's equations in vacuum with a positive cosmological constant which can be considered as inhomogeneous generalizations of the Nariai cosmological model. Similar to this Nariai spacetime, our solutions ... More

SO*(2N) coherent states for loop quantum gravityJan 25 2017A SU(2) intertwiner with N legs can be interpreted as the quantum state of a convex polyhedron with N faces (when working in 3d). We show that the intertwiner Hilbert space carries a representation of the non-compact group SO*(2N). This group can be viewed ... More

Classification of digital affine noncommutative geometriesJan 22 2017It is known that connected translation invariant $n$-dimensional noncommutative differentials $d x^i$ on the algebra $k[x^1,\cdots,x^n]$ of polynomials in $n$-variables over a field $k$ are classified by commutative algebras $V$ on the vector space spanned ... More

Topological entropy for locally linearly compact vector spacesJan 03 2017In analogy to the topological entropy for continuous endomorphisms of totally disconnected locally compact groups, we introduce a notion of topological entropy for continuous endomorphisms of locally linearly compact vector spaces. We study the fundamental ... More