Related papers: Metrically universal abelian groups

We give a classification and complete algebraic description of groups allowing only finitely many (left multiplication invariant) circular orders. In particular, they are all solvable groups with a specific semi-direct product…

A metric space is indivisible if for any partition of it into finitely many pieces one piece contains an isometric copy of the whole space. Continuing our investigation of indivisible metric spaces, we show that a countable ultrametric…

In this paper we begin the systematic study of group equations with abelian predicates in the main classes of groups where solving equations is possible. We extend the line of work on word equations with length constraints, and more…

A topological space $X$ is defined to have a neighborhood $P$-base at any $x\in X$ from some poset $P$ if there exists a neighborhood base $(U_p[x])_{p\in P}$ at $x$ such that $U_p[x]\subseteq U_{p'}[x]$ for all $p\geq p'$ in $P$. We prove…

Let M be a surface (possibly nonorientable) with punctures and/or boundary components. The paper is a study of ``geometric subgroups'' of the mapping class group of M, that is subgroups corresponding to inclusions of subsurfaces (possibly…

This work addresses the questions: (i) Among all left-invariant Riemannian metrics on a given Lie group, is there any whose isometry group or isometry algebra contain that of all others? (ii) Do expanding left-invariant Ricci solitons…

Each Abelian subgroup of the fundamental group of a compact and locally simply connected $d$-dimensional length space with no conjugate points is isomorphic to $\mathbb{Z}^k$ for some $0 \leq k \leq d$. It follows from this and previously…

We define a metric ultraproduct of topological groups with left-invariant metric, and show that there is a countable sequence of finite groups with left-invariant metric whose metric ultraproduct contains isometrically as a subgroup every…

In this article we prove some previously announced results about metric ultraproducts of finite simple groups. We show that any non-discrete metric ultraproduct of alternating or special linear groups is a geodesic metric space. For more…

A characteristic result is that if 2^{aleph_0}< mu < mu^+< lambda = cf(lambda)< mu^{aleph_0}, then among the separable reduced p-groups of cardinality lambda which are (< lambda)-stable there is no universal one.

In this work we investigate solvable and nilpotent Lie groups with special metrics. The metrics of interest are left-invariant Einstein and algebraic Ricci soliton metrics. Our main result shows that the existence of a such a metric is…

We give a general procedure for constructing metric spaces from systems of partitions. This generalises and provides analogues of Sageev's construction of dual CAT(0) cube complexes for the settings of hyperbolic and injective metric…

The distribution of cardinalities of zero-sum sets in abelian groups is completely determined. A complex summation involving the M\"obius function is given for the general abelian group, while in many special cases, including the case of…

We study invariant and bi-invariant metrics on groups focusing on finite groups $G$. We show that non-equivalent (bi) invariant metrics on $G$ are in 1-1 correspondence with unitary symmetric (conjugate) partitions on $G$. To every metric…

A group is Artinian if there is no infinite strictly descending chain of subgroups. Ol'shanskii has asked whether there are Artinian groups of arbitrarily large cardinality. We show that this problem is essentially the same as an analogous…

For a finite group $G$ and $U: = U(\mathbb{Z}G)$, the group of units of the integral group ring of $G$, we study the implications of the structure of $G$ on the abelianization $U/U'$ of $U$. We pose questions on the connections between the…

We give a short and simple proof, utilizing the pre-determinant of P. de la Harpe and G. Skandalis, that the universal covering group of the unitary group of a II$_1$ von Neumann algebra $\mathcal{M}$, when equipped with the norm topology,…

In this paper, we prove a series of results on group embeddings in groups with a small number of generators. We show that each finitely generated group $G$ lying in a variety ${\mathcal M}$ can be embedded in a $4$-generated group $H \in…

In this paper we give invariants that characterize isotypically equivalent Abelian periodic groups. Also, we describe types of standart tuples of elements in these groups. As the particular case we prove that two Abelian $p$-groups with…

A metric space $\mathrm{M}=(M,\de)$ is {\em indivisible} if for every colouring $\chi: M\to 2$ there exists $i\in 2$ and a copy $\mathrm{N}=(N, \de)$ of $\mathrm{M}$ in $\mathrm{M}$ so that $\chi(x)=i$ for all $x\in N$. The metric space…