Narrow Results

In this paper, we compute the $K$- and $KO$-theory for the classifying $G$-spaces $\underline{E}G$ for proper actions of certain infinite discrete groups $G$ via the equivariant Atiyah–Hirzebruch spectral sequence.

We describe a class of non-linear transformations acting on many variables. These transformations have their origin in the theory of quantum integrability: they appear in the description of the symmetries of the Yang-Baxter equations and their higher dimensional generalizations. They are generated by involutions and act as birational mappings on various projective spaces. We present numerous figures, showing successive iterations of these mappings. The existence of algebraic invariants explains the aspect of these figures. We also study deformations of our transformations.

This is the final paper in a series of four, concerning the surface 𝕋 × 𝕋 embedded in ℂℙ⁸, where 𝕋 is the one-dimensional torus. In this paper we compute the fundamental group of the Galois cover of the surface with respect to a generic projection onto ℂℙ², and show that it is nilpotent of class 3. This is the first time such a group is presented as the fundamental group of a Galois cover of a surface.

We explicitly construct Markov languages of normal forms for the groups in the title of the paper and closely related groups. A Markov language of normal forms is a choice of "preferred spelling" for each group element such that the collection of choices is particularly simple in a language theoretic sense.

A generalization of the Kauffman tangle algebra is given for Coxeter type D_n. The tangles involve a pole of order 2. The algebra is shown to be isomorphic to the Birman–Murakami–Wenzl algebra of the same type. This result extends the isomorphism between the two algebras in the classical case, which, in our set-up, occurs when the Coxeter type is A_{n - 1}. The proof involves a diagrammatic version of the Brauer algebra of type D_n of which the generalized Temperley–Lieb algebra of type D_n is a subalgebra.

We introduce a natural $G_n^k$-type construction of groups on any given simplicial complex $K$ called $G^k(K)$ as a generalization of the notion of groups $G_n^k$, which gives a functor from the category of simplicial complexes and injective simplicial maps to the category of groups. We prove that the word problem on $G^2(K)$ is solvable.

Let L be an unsplittable link in S³ and let V denote the double cover of S³ branched along L. Starting with an arbitrary planar projection of L, it is shown how to construct a special presentation for the fundamental group π₁V which encodes in a simple way geometric data from a (checkerboard) spanning surface obtained from the given projection, and which relates to the construction of Coxeter quotients of π₁ (S³-L) previously studied by the author and Y. W. Lee.

We study the wavelet transform of functions invariant under a symmetry group, where the wavelet transform is associated to an irreducible unitary group representation. Among other results a new inversion formula and a new covariance principle are derived. As main examples we discuss the continuous wavelet transform and the short time Fourier transform of radially symmetric functions on ℝ^d.

Four-dimensional A₄ polytopes and their dual polytopes have been constructed as the orbits of the Coxeter–Weyl group W(A₄) where the group elements and the vertices of the polytopes are represented by quaternions. Projection of an arbitrary W(A₄) orbit into three dimensions is made using the subgroup W(A₃). A generalization of the Catalan solids for 3D-polyhedra has been developed and dual polytopes of the uniform A₄ polytopes have been constructed.

Four-dimensional F₄ polytopes and their dual polytopes have been constructed as the orbits of the Coxeter–Weyl group W(F₄ ) where the group elements and the vertices of the polytopes are represented by quaternions. Branchings of an arbitrary W(F₄ ) orbit under the Coxeter groups W(B₄) and W(B₃) × W(A₁) have been presented. The role of group theoretical technique and the use of quaternions have been emphasized.

We associate the lepton–quark families with the vertices of the 4D polytopes 5-cell ${(0001)}_{A4}$ and the rectified 5-cell ${(0100)}_{A4}$ derived from the $SU(5)$ Coxeter–Dynkin diagram. The off-diagonal gauge bosons are associated with the root polytope ${(1001)}_{A4}$ whose facets are tetrahedra and the triangular prisms. The edge-vertex relations are interpreted as the $SU(5)$ charge conservation. The Dynkin diagram symmetry of the $SU(5)$ diagram can be interpreted as a kind of particle-antiparticle symmetry. The Voronoi cell of the root lattice consists of the union of the polytopes ${{{{(1000)}_{A4}+(0100)}_{A4}+(0010)}_{A4}+(0001)}_{A4}$ whose facets are 20 rhombohedra. We construct the Delone (Delaunay) cells of the root lattice as the alternating 5-cell and the rectified 5-cell, a kind of dual to the Voronoi cell. The vertices of the Delone cells closest to the origin consist of the root vectors representing the gauge bosons. The faces of the rhombohedra project onto the Coxeter plane as thick and thin rhombs leading to Penrose-like tiling of the plane which can be used for the description of the 5-fold symmetric quasicrystallography. The model can be extended to $SO(10)$ and even to $SO(11)$ by noting the Coxeter–Dynkin diagram embedding $A_4\subset D_5\subset B_5$. Another embedding can be made through the relation $A_4\subset D_5\subset E_6$ for more popular $GU{T}^{\prime }s$. Appendix A includes the quaternionic representations of the Coxeter–Weyl groups $W(A_4)\subset {W(H}_4)$ which can be obtained directly from ${W(E}_8)$ by projection. This leads to relations of the $SU(5)$ polytopes with the quasicrystallography in 4D and $E_8$ polytopes. Appendix B discusses the branching of the polytopes in terms of the irreducible representations of the Coxeter–Weyl group $W(A_4)\approx S_5$.

Recently, the first named author defined a 2-parametric family of groups $G_n^k$ [V. O. Manturov, Non–reidemeister knot theory and its applications in dynamical systems, geometry and topology, preprint (2015), arXiv:1501.05208]. Those groups may be regarded as analogues of braid groups.

Study of the connection between the groups $G_n^k$ and dynamical systems led to the discovery of the following fundamental principle: “If dynamical systems describing the motion of $n$ particles possess a nice codimension one property governed by exactly $k$ particles, then these dynamical systems admit a topological invariant valued in $G_n^k$”.

The $G_n^k$ groups have connections to different algebraic structures, Coxeter groups, Kirillov-Fomin algebras, and cluster algebras, to name three. Study of the $G_n^k$ groups led to, in particular, the construction of invariants, valued in free products of cyclic groups. All generators of the $G_n^k$ groups are reflections which make them similar to Coxeter groups and not to braid groups. Nevertheless, there are many ways to enhance $G_n^k$ groups to get rid of this $2$-torsion.

Later the first and the fourth named authors introduced and studied the second family of groups, denoted by ${\mathrm{\Gamma }}_n^k$, which are closely related to triangulations of manifolds.

The spaces of triangulations of a given manifolds have been widely studied. The celebrated theorem of Pachner [P.L. homeomorphic manifolds are equivalent by elementary shellings, Europ. J. Combin.12(2) (1991) 129–145] says that any two triangulations of a given manifold can be connected by a sequence of bistellar moves or Pachner moves. See also [I. M. Gelfand, M. M. Kapranov and A. V. Zelevinsky, Discriminants, Resultants, and Multidimensional Determinants (Birkhäuser, Boston, 1994); A. Nabutovsky, Fundamental group and contractible closed geodesics, Comm. Pure Appl. Math.49(12) (1996) 1257–1270]; the ${\mathrm{\Gamma }}_n^k$ naturally appear when considering the set of triangulations with the fixed number of points.

There are two ways of introducing the groups ${\mathrm{\Gamma }}_n^k$: the geometrical one, which depends on the metric, and the topological one. The second one can be thought of as a “braid group” of the manifold and, by definition, is an invariant of the topological type of manifold; in a similar way, one can construct the smooth version.

In this paper, we give a survey of the ideas lying in the foundation of the $G_n^k$ and ${\mathrm{\Gamma }}_n^k$ theories and give an overview of recent results in the study of those groups, manifolds, dynamical systems, knot and braid theories.

We study relatively hyperbolic Coxeter groups of type $HM$ with maximal affine Coxeter subgroups of codimension one. Our main result in this paper is that the dimension of these groups is bounded above.