PROPER AFFINE DEFORMATIONS OF THE ONE-HOLED TORUS

A Margulis spacetime is a complete at Lorentzian 3-manifold M with free fundamental group. Associated to M is a noncompact complete hyperbolic surface ∑ homotopy-equivalent to M. The purpose of this paper is to classify Margulis spacetimes when ∑ is homeomorphic to a one-holed torus. We show that every such M decomposes into polyhedra bounded by crooked planes, corresponding to an ideal triangulation of ∑. This paper classifies and analyzes the structure of crooked ideal triangles, which play the same role for Margulis spacetimes as ideal triangles play for hyperbolic surfaces.     

Contact open books with exotic pages

We consider a fixed contact 3-manifold that admits infinitely many compact Stein fillings which are all homeomorphic but pairwise non-diffeomorphic. Each of these fillings gives rise to a closed contact 5-manifold described as a contact open book whose page is the filling at hand and whose monodromy is the identity symplectomorphism. We show that the resulting infinitely many contact 5-manifolds are all diffeomorphic but pairwise non-contactomorphic. Moreover, we explicitly determine these contact 5-manifolds.     

QUANDLE VARIETIES,GENERALIZED SYMMETRIC SPACES,AND <Emphasis Type="Italic">φ</Emphasis>-SPACES

We define a quandle variety as an irreducible algebraic variety Q endowed with an algebraically defined quandle operation ?. It can also be seen as an analogue of a generalized affine symmetric space or a regular s-manifold in algebraic geometry.Assume that Q is normal as an algebraic variety and that the action of its inner automorphism group Inn(Q) has a dense orbit. Then we show that there is an algebraic group G acting on Q with the same orbits as Inn(Q) such that each G-orbit is isomorphic to the quandle (G/H, ?φ) associated to the group G, an automorphism φ of G and a subgroup H of Gφ.     

Positive-fraction intersection results and variations of weak epsilon-nets

A connected Finsler space (M, F) is said to be homogeneous if it admits a transitive connected Lie group G of isometries. A geodesic in a homogeneous Finsler space (G / H, F) is called a homogeneous geodesic if it is an orbit of a one-parameter subgroup of G. In this paper, we study the problem of the existence of homogeneous geodesics on a homogeneous Finsler space, and prove that any homogeneous Finsler space of odd dimension admits at least one homogeneous geodesic through each point.     

A perfect non-topologizable group

The first example of a group G whose automorphism group Aut (G) admits only discrete separated topology is presented in the paper. The group G is isomorphic to Aut (G) and all elements of the group G except for the unity satisfy some equation w(x) = 1.     

Approximating <Emphasis Type="Italic">C</Emphasis><Superscript>0</Superscript>-foliations by contact structures

We show that any co-orientable foliation of dimension two on a closed orientable 3-manifold with continuous tangent plane field can be C ⁰-approximated by both positive and negative contact structures unless all leaves of the foliation are simply connected. As applications we deduce that the existence of a taut C ⁰-foliation implies the existence of universally tight contact structures in the same homotopy class of plane fields and that a closed 3-manifold that admits a taut C ⁰-foliation of codimension-1 is not an L-space in the sense of Heegaard-Floer homology.     

On the rank of compact <Emphasis Type="Italic">p</Emphasis>-adic Lie groups

The rank of a profinite group G is the basic invariant \({{\rm rk}(G):={\rm sup}\{d(H) \mid H \leq G\}}\), where H ranges over all closed subgroups of G and d(H) denotes the minimal cardinality of a topological generating set for H. A compact topological group G admits the structure of a p-adic Lie group if and only if it contains an open pro-p subgroup of finite rank. For every compact p-adic Lie group G one has rk(G) ≥ dim(G), where dim(G) denotes the dimension of G as a p-adic manifold. In this paper we consider the converse problem, bounding rk(G) in terms of dim(G). Every profinite group G of finite rank admits a maximal finite normal subgroup, its periodic radical π(G). One of our main results is the following. Let G be a compact p-adic Lie group such that π(G) = 1, and suppose that p is odd. If \(\{g \in G \mid g^{p-1}=1 \}\) is equal to {1}, then rk(G) = dim(G).     

A solvability criterion for finite groups

We show that if for every prime p, the normalizer of a Sylow p-subgroup of a finite group G admits a p-solvable supplement, then G is solvable. This generalizes a solvability criterion of Hall which asserts that a finite group G is solvable if and only if G has a Hall p′-subgroup for every prime p.     

On a question of Külshammer for representations of finite groups in reductive groups

Let G be a simple algebraic group of type G₂ over an algebraically closed field of characteristic 2. We give an example of a finite group Γ with Sylow 2-subgroup Γ₂ and an infinite family of pairwise non-conjugate homomorphisms ρ: Γ → G whose restrictions to Γ₂ are all conjugate. This answers a question of Burkhard Külshammer from 1995. We also give an action of Γ on a connected unipotent group V such that the map of 1-cohomologies H¹(Γ, V) → H¹(Γ_p, V) induced by restriction of 1-cocycles has an infinite fibre.     

Sensitive open maps on Peano continua having a free arc

Let X be a Peano continuum having a free arc. If X admits a sensitive open map, then X either is homeomorphic to the closed interval [0, 1], or is homeomorphic to the unit circle S¹.     

Equivariant bundles and connections

Let X be a connected complex manifold equipped with a holomorphic action of a complex Lie group G. We investigate conditions under which a principal bundle on X admits a G-equivariance structure.     

Group distance magic and antimagic graphs

Given a graph G with n vertices and an Abelian group A of order n, an A-distance antimagic labelling of G is a bijection from V (G) to A such that the vertices of G have pairwise distinct weights, where the weight of a vertex is the sum (under the operation of A) of the labels assigned to its neighbours. An A-distance magic labelling of G is a bijection from V (G) to A such that the weights of all vertices of G are equal to the same element of A. In this paper we study these new labellings under a general setting with a focus on product graphs. We prove among other things several general results on group antimagic or magic labellings for Cartesian, direct and strong products of graphs. As applications we obtain several families of graphs admitting group distance antimagic or magic labellings with respect to elementary Abelian groups, cyclic groups or direct products of such groups.     

Some Triangulated Surfaces without Balanced Splitting

Let G be the graph of a triangulated surface \(\varSigma \) of genus \(g\ge 2\). A cycle of G is splitting if it cuts \(\varSigma \) into two components, neither of which is homeomorphic to a disk. A splitting cycle has type k if the corresponding components have genera k and \(g-k\). It was conjectured that G contains a splitting cycle (Barnette 1982). We confirm this conjecture for an infinite family of triangulations by complete graphs but give counter-examples to a stronger conjecture (Mohar and Thomassen in Graphs on surfaces. Johns Hopkins studies in the mathematical sciences. Johns Hopkins University Press, Baltimore, 2001) claiming that G should contain splitting cycles of every possible type.     

<Emphasis Type="Italic">r</Emphasis>-Dynamic Chromatic Number of Some Line Graphs

An r-dynamic coloring of a graph G is a proper coloring c of the vertices such that |c(N(v))| ≥ min {r, deg(v)}, for each v ∈ V (G). The r-dynamic chromatic number of a graph G is the smallest k such that G admits an r-dynamic coloring with k colors. In this paper, we obtain the r-dynamic chromatic number of the line graph of helm graphs H_n for all r between minimum and maximum degree of H_n. Moreover, our proofs are constructive, what means that we give also polynomial time algorithms for the appropriate coloring. Finally, as the first, we define an equivalent model for edge coloring.     

Ergodic theorems for coset spaces

We study in this paper the validity of the Mean Ergodic Theorem along left Følner sequences in a countable amenable group G. Although the Weak Ergodic Theorem always holds along any left Følner sequence in G, we provide examples where the Mean Ergodic Theorem fails in quite dramatic ways. On the other hand, if G does not admit any ICC quotients, e.g., if G is virtually nilpotent, then the Mean Ergodic Theorem holds along any left Følner sequence. In the case when a unitary representation of a countable amenable group is induced from a unitary representation of a “sufficiently thin” subgroup, we show that the Mean Ergodic Theorem holds for the induced representation along any left Følner sequence. Furthermore, we show that every countable (infinite) amenable group L embeds into a countable (not necessarily amenable) group G which admits a unitary representation with the property that for any left Følner sequence (F_n) in L, there exists a sequence (s_n) in G such that the Mean (but not the Weak) Ergodic Theorem fails in a rather strong sense along the (right-translated) sequence (F_ns_n) in G. Finally, we provide examples of countable (not necessarily amenable) groups G with proper, infinite-index subgroups H, so that the Pointwise Ergodic Theorem holds for averages along any strictly increasing and nested sequence of finite subsets of the coset G/H.     

Residually finite algorithmically finite groups,their subgroups and direct products

We construct a finitely generated infinite recursively presented residually finite algorithmically finite group G, thus answering a question of Myasnikov and Osin. The group G here is “strongly infinite” and “strongly algorithmically finite,” which means that G contains an infinite Abelian normal subgroup and all finite Cartesian powers of G are algorithmically finite (i.e., for any n, there is no algorithm writing out infinitely many pairwise distinct elements of the group G ⁿ ). We also formulate several open questions concerning this topic.     

Judicious bisection of hypergraphs

Judicious bisection of hypergraphs asks for a balanced bipartition of the vertex set that optimizes several quantities simultaneously.In this paper,we prove that if G is a hypergraph with n vertices and m_i edges of size i for i=1,2,...,k,then G admits a bisection in which each vertex class spans at most(m_1)/2+1/4m_2+…+(1/(2~k)+m_k+o(m_1+…+m_k) edges,where G is dense enough or △(G)= o(n) but has no isolated vertex,which turns out to be a bisection version of a conjecture proposed by Bollobas and Scott.     

Orbit projections as fibrations

The orbit projection π: M → M/G of a proper G-manifold M is a fibration if and only if all points in M are regular. Under additional assumptions we show that π is a quasifibration if and only if all points are regular. We get a full answer in the equivariant category: π is a G-quasifibration if and only if all points are regular.     

ON ALGEBRAIC VOLUME DENSITY PROPERTY

A smooth affine algebraic variety X equipped with an algebraic volume form ω has the algebraic volume density property (AVDP) if the Lie algebra generated by complete algebraic vector fields of ω-divergence zero coincides with the space of all algebraic vector fields of ω-divergence zero. We develop an effective criterion of verifying whether a given X has AVDP. As an application of this method we establish AVDP for any homogeneous space X = G/R that admits a G-invariant algebraic volume form where G is a linear algebraic group and R is a closed reductive subgroup of G.     

On dihedrants admitting arc-regular group actions

We consider Cayley graphs Γ over dihedral groups, dihedrants for short, which admit an automorphism group G acting regularly on the arc set of Γ. We prove that, if D _2n ≤G≤Aut(Γ) is a regular dihedral subgroup of G of order 2n such that any of its index 2 cyclic subgroups is core-free in G, then Γ belongs to the family of graphs of the form \((K_{n_{1}}\otimes\cdots\otimes K_{n_{\ell}})[K_{m}^{\mathrm{c}}]\), where 2n=n ₁???n _? m, and the numbers n _i are pairwise coprime. Applications to 1-regular dihedrants and Cayley maps on dihedral groups are also given.