Bour’s theorem and helicoidal surfaces with constant mean curvature in the Bianchi–Cartan–Vranceanu spaces

In this paper, we generalize a classical result of Bour concerning helicoidal surfaces in the three-dimensional Euclidean space $${\mathbb {R}}^3$$ R 3 to the case of helicoidal surfaces in the Bianchi–Cartan–Vranceanu (BCV) spaces, i.e., in the Riemannian 3-manifolds whose metrics have groups of isometries of dimension 4 or 6, except the hyperbolic one. In particular, we prove that in a BCV-space there exists a two-parameter family of helicoidal surfaces isometric to a given helicoidal surface; then, by making use of this two-parameter representation, we characterize helicoidal surfaces which have constant mean curvature, including the minimal ones.

AUTOISOMETRIES AND AUTOSIMILARITIES OF LIE ALGEBRA 𝒜(1) ⊕ ℛ2

We consider four-dimensional Lie algebra 𝒜(1) ⊕ ℛ2 endowed with Lorentzian scalar product. We find all the one-parameter groups of isometries and similarities, which are simultaneously automorphisms of Lie algebra, and also we find the conditions of existence of such one-parameter group. Conditions of existence are associated with the location of ideals with respect to isotropic cone.

ORDINALITY OF ISOMETRY GROUPS OF HAMMING SPACES OF PERIODIC SEQUENCES

We study Hamming spaces (known also as measure algebras). For all Steinitz numbers s , we find cardinalities of the groups of isometries of Hamming spaces of s -periodic sequences and the group of automorphisms of such space and we prove that are both cardinalities equal to Pic 1.

Determination of the Homology and the Cohomology of a Few Groups of Isometries of the Hyperbolic Plane

In this paper we determine the homology and the cohomology groups of two properly discontinuous groups of isometries of the hyperbolic plane having non-compact orbit spaces and the fundamental group of a graph of groups with a finite vertex groups and no trivial edges by extending Lyndon’s partial free resolution for finitely presented groups. For the first two groups, we obtain partial extensions and the corresponding homology. We also compute the corresponding cohomology groups for one of these groups. Finally we obtain homology and cohomology in all dimensions for the last of the above mentioned groups by constructing a full resolution for this group.GANIT J. Bangladesh Math. Soc.Vol. 36 (2016) 65-77

AUTOMATIC CONTINUITY FOR ISOMETRY GROUPS

We present a general framework for automatic continuity results for groups of isometries of metric spaces. In particular, we prove automatic continuity property for the groups of isometries of the Urysohn space and the Urysohn sphere, i.e. that any homomorphism from either of these groups into a separable group is continuous. This answers a question of Ben Yaacov, Berenstein and Melleray. As a consequence, we get that the group of isometries of the Urysohn space has unique Polish group topology and the group of isometries of the Urysohn sphere has unique separable group topology. Moreover, as an application of our framework we obtain new proofs of the automatic continuity property for the group $\text{Aut}([0,1],\unicode[STIX]{x1D706})$, due to Ben Yaacov, Berenstein and Melleray and for the unitary group of the infinite-dimensional separable Hilbert space, due to Tsankov.