Cyclic conjugacy separability and conjugacy separability of certain HNN extensions

Abstract

In this note, we give a criterion for certain HNN extensions of cyclic conjugacy separable (respectively conjugacy separable) groups with infinite cyclic associated subgroups to be again cyclic conjugacy separable (respectively conjugacy separable).

Communicated by Alexander Olshanskii     

Computing circular separability

Two sets of planar pointsS ₁ andS ₂ are circularly separable if there is a circle that enclosesS ₁ but excludesS ₂. We show that deciding whether two sets are circularly separable can be accomplished inO(n) time using linear programming. We also show that a smallest separating circle can be found inO(n) time, and largest separating circles can be found inO(n logn) time. Finally we establish that all these results are optimal.     

Conjugacy separability of Seifert 3-manifold groups over non-orientable surfaces

Scott (1978) [12] showed Seifert 3-manifold groups are subgroup separable. Niblo (1992) [9] improved this result by showing that these groups are double coset separable. In Allenby, Kim and Tang (2005) [2] it was shown that all but two types of groups in the orientable case are conjugacy separable. Martino (2007) [7] using topological results showed that Seifert groups are conjugacy separable. Here we use algebraic method to show that Seifert groups over non-orientable surfaces are conjugacy separable.     

Selective separability and its variations

A space X is said to be selectively separable (=M-separable) if for each sequence {Dn:n∈ω} of dense subsets of X, there are finite sets Fn⊂Dn (n∈ω) such that ?{Fn:n∈ω} is dense in X. On selective separability and its variations, we show the following: (1) Selective separability, R-separability and GN-separability are preserved under finite unions; (2) Assuming CH (the continuum hypothesis), there is a countable regular maximal R-separable space X such that X² is not selectively separable; (3) c{0,1} has a selectively separable, countable and dense subset S such that the group generated by S is not selectively separable. These answer some questions posed in Bella et al. (2008) [7].     

Conjugacy separability of 1-acylindrical graphs of free groups

We use the theory of group actions on profinite trees to prove that the fundamental group of a finite, 1-acylindrical graph of free groups with finitely generated edge groups is conjugacy separable. This has several applications: we prove that positive, C′(1/6) one-relator groups are conjugacy separable; we provide a conjugacy separable version of the Rips construction; we use this latter to provide an example of two finitely presented, residually finite groups that have isomorphic profinite completions, such that one is conjugacy separable and the other does not even have solvable conjugacy problem.     

A successive approximation method for quantum separability

Determining whether a quantum state is separable or inseparable （entangled） is a problem of fundamental importance in quantum science and has attracted much attention since its first recognition by Einstein, Podolsky and Rosen [Phys. Rev., 1935, 47： 777] and SchrSdinger [Naturwissenschaften, 1935, 23： 807-812, 823-828, 844-849]. In this paper, we propose a successive approximation method （SAM） for this problem, which approximates a given quantum state by a so-called separable state： if the given states is separable, this method finds its rank-one components and the associated weights; otherwise, this method finds the distance between the given state to the set of separable states, which gives information about the degree of entanglement in the system. The key task per iteration is to find a feasible descent direction, which is equivalent to finding the largest M-eigenvalue of a fourth-order tensor. We give a direct method for this problem when the dimension of the tensor is 2 and a heuristic cross-hill method for cases of high dimension. Some numerical results and experiences are presented.     

Borel functions and separability of metric spaces

Let X and Y be metric spaces with X separable, and let \(f: X\rightarrow Y\) be a Borel function. Is then f(X) separable? In this paper, we prove that this problem is independent of ZFC. We also give a partial answer to an open problem which was asked by A. H. Stone.     

Proper forcing axiom and selective separability

We continue the study of Selectively Separable (SS) and, a game-theoretic strengthening, strategically selectively separable spaces (SS⁺) (see Barman, Dow (2011) [1]). The motivation for studying SS⁺ is that it is a property possessed by all separable subsets of Cp(X) for each σ-compact space X. We prove that the winning strategy for countable SS⁺ spaces can be chosen to be Markov. We introduce the notion of being compactlike for a collection of open sets in a topological space and with the help of this notion we prove that there are two countable SS⁺ spaces such that the union fails to be SS⁺, which contrasts the known result about SS spaces. We also prove that the product of two countable SS⁺ spaces is again countable SS⁺. One of the main results in this paper is that the proper forcing axiom, PFA, implies that the product of two countable Fréchet spaces is SS, a statement that was shown in Barman, Dow (2011) [1] to consistently fail. An auxiliary result is that it is consistent with the negation of CH that all separable Fréchet spaces have π-weight at most ω₁.     

Final compactness and separability in regular symmetrizable spaces

In 1966 A. V. Arkhangel'skii posed the following question: Is it true that every regular finally compact symmetrizable space is separable? S. I. Nedev soon showed that a regular finally compact symmetrizable space is hereditarily finally compact. Consequently any counterexample to Arkhangel'skii's conjecture must be an L-space. Applying the technique of iterated forcing we prove that in the axiom systemZFC for set theory it is consistent to assume the existence of a regular (hereditarily) finally compact symmetrizable space X that is nonseparable. Thus it is impossible to prove using the axiom systemZFC that every regular finally compact symmetrizable space is separable. The space X has additional properties as well: it has a basis consisting of open/closed sets (i.e., it is zero-dimensional in the sense ofind, it can be mapped continuously and one-to-one onto a separable metric space, it is α-left and has cardinality ω₁. Bibliography: 25 titles.     

On connection between the switching separability of a graph and its subgraphs

A graph of order n ≥ 4 is called switching separable if its modulo-2 sum with some complete bipartite graph on the same set of vertices is divided into two mutually independent subgraphs, each having at least two vertices. We prove the following: If removing any one or two vertices of a graph always results in a switching separable subgraph then the graph itself is switching separable. On the other hand, for each odd order greater than 4, there is a graph that is not switching separable, but removing a vertex always results in a switching separable subgraph. We show some connection with similar facts on the separability of Boolean functions and the reducibility of n-ary quasigroups.     

Abelian subgroup separability of certain generalized free products of free or finitely generated nilpotent groups

In this paper we prove that certain generalized free products of abelian subgroup separable groups, amalgamating an infinite cyclic subgroup, are abelian subgroup separable. Applying this, we derive that tree products of free groups or finitely generated nilpotent groups, amalgamating infinite cyclic subgroups, are abelian subgroup separable.     

Sequential + separable vs sequentially separable and another variation on selective separability

A space X is sequentially separable if there is a countable D ? X such that every point of X is the limit of a sequence of points from D. Neither “sequential + separable” nor “sequentially separable” implies the other. Some examples of this are presented and some conditions under which one of the two implies the other are discussed. A selective version of sequential separability is also considered.     

Subgroup separability in residually free groups

We prove that the finitely presentable subgroups of residually free groups are separable and that the subgroups of type are virtual retracts. We describe a uniform solution to the membership problem for finitely presentable subgroups of residually free groups.     

Virtual retractions,conjugacy separability and omnipotence

We use wreath products to provide criteria for a group to be conjugacy separable or omnipotent. These criteria are in terms of virtual retractions onto cyclic subgroups. We give two applications: a straightforward topological proof of the theorem of Stebe that infinite-order elements of Fuchsian groups (of the first type) are conjugacy distinguished, and a proof that surface groups are omnipotent.     

A likelihood ratio test for separability of covariances

We propose a formal test of separability of covariance models based on a likelihood ratio statistic. The test is developed in the context of multivariate repeated measures (for example, several variables measured at multiple times on many subjects), but can also apply to a replicated spatio-temporal process and to problems in meteorology, where horizontal and vertical covariances are often assumed to be separable. Separable models are a common way to model spatio-temporal covariances because of the computational benefits resulting from the joint space-time covariance being factored into the product of a covariance function that depends only on space and a covariance function that depends only on time. We show that when the null hypothesis of separability holds, the distribution of the test statistic does not depend on the type of separable model. Thus, it is possible to develop reference distributions of the test statistic under the null hypothesis. These distributions are used to evaluate the power of the test for certain nonseparable models. The test does not require second-order stationarity, isotropy, or specification of a covariance model. We apply the test to a multivariate repeated measures problem.     

Subgroup separability of certain generalized free products of nilpotent-by-finite groups

In this paper, we show that certain generalized free products of nilpotent-by-finite groups are subgroup separable when the amalgamated subgroup is h× D where D is in the center of both factors.     

Testing separability in marked multidimensional point processes with covariates

In modeling marked point processes, it is convenient to assume a separable or multiplicative form for the conditional intensity, as this assumption typically allows one to estimate each component of the model individually. Tests have been proposed in the simple marked point process case, to investigate whether the mark distribution is separable from the spatial–temporal characteristics of the point process. Here, we extend these tests to the case of a marked point process with covariates, and where one is interested in testing the separability of each of the covariates, as well as the mark and the coordinates of the point process. The extension is not at all trivial, and covariates must be treated in a fundamentally different way than marks and coordinates of the process, especially when the covariates are not uniformly distributed. An application is given to point process models for forecasting wildfire hazard in Los Angeles County, California, and solutions are proposed to the problem of how to proceed when the separability hypothesis is rejected.     

Subgroup separability, knot groups and graph manifolds

This paper answers a question of Burns, Karrass and Solitar by giving examples of knot and link groups which are not subgroup-separable. For instance, it is shown that the fundamental group of the square knot complement is not subgroup separable. Let denote the fundamental group of the link consisting of a chain of circles. It is shown that is not subgroup separable. Furthermore, it is shown that is a subgroup of every known non-subgroup separable compact 3-manifold group. It is asked whether all such examples contain .

     

Finite dimensionality and separability of the stalks of Banach bundles

Topological characteristics are studied of the set of points at which the stalks of an ample Banach bundle over an extremally disconnected compact space are finite-dimensional or separable. Some connection is established between finite dimensionality or separability of the stalks of a bundle and the analogous properties of the stalks of the ample hull of the bundle. We obtain a new criterion for existence of a dual bundle.     

Variations of selective separability

A space X is selectively separable if for every sequence of dense subspaces of X one can select finite Fn⊂Dn so that is dense in X. In this paper selective separability and variations of this property are considered in two special cases: Cp spaces and dense countable subspaces in κ².