scholarly journals Uniformly perfect finitely generated simple left orderable groups

2019 ◽  
Vol 41 (2) ◽  
pp. 534-552
Author(s):  
JAMES HYDE ◽  
YASH LODHA ◽  
ANDRÉS NAVAS ◽  
CRISTÓBAL RIVAS
Keyword(s):  
Fixed Point ◽  
Real Line ◽  
Piecewise Linear ◽  
Simple Groups ◽  
Finitely Generated ◽  
The Real ◽  
Link Type ◽  
Open Questions ◽  
Uniformly Perfect

We show that the finitely generated simple left orderable groups $G_{\!\unicode[STIX]{x1D70C}}$ constructed by the first two authors in Hyde and Lodha [Finitely generated infinite simple groups of homeomorphisms of the real line. Invent. Math. (2019), doi:10.1007/s00222-019-00880-7] are uniformly perfect—each element in the group can be expressed as a product of three commutators of elements in the group. This implies that the group does not admit any homogeneous quasimorphism. Moreover, any non-trivial action of the group on the circle, which lifts to an action on the real line, admits a global fixed point. It follows that any faithful action on the real line without a global fixed point is globally contracting. This answers Question 4 of the third author [A. Navas. Group actions on 1-manifolds: a list of very concrete open questions. Proceedings of the International Congress of Mathematicians, Vol. 2. Eds. B. Sirakov, P. Ney de Souza and M. Viana. World Scientific, Singapore, 2018, pp, 2029–2056], which asks whether such a group exists. This question has also been answered simultaneously and independently, using completely different methods, by Matte Bon and Triestino [Groups of piecewise linear homeomorphisms of flows. Preprint, 2018, arXiv:1811.12256]. To prove our results, we provide a characterization of elements of the group $G_{\!\unicode[STIX]{x1D70C}}$ which is a useful new tool in the study of these examples.

scholarly journals Examples of expanding maps with some special properties

1987 ◽  
Vol 36 (3) ◽  
pp. 469-474 ◽  
Author(s):  
Bau-Sen Du
Keyword(s):  
Fixed Point ◽  
Positive Integer ◽  
Periodic Point ◽  
Topological Entropy ◽  
Real Line ◽  
Periodic Points ◽  
Unit Interval ◽  
Expanding Maps ◽  
The Real ◽  
Piecewise Monotonic

Let I be the unit interval [0, 1] of the real line. For integers k ≥ 1 and n ≥ 2, we construct simple piecewise monotonic expanding maps Fk, n in C0 (I, I) with the following three properties: (1) The positive integer n is an expanding constant for Fk, n for all k; (2) The topological entropy of Fk, n is greater than or equal to log n for all k; (3) Fk, n has periodic points of least period 2k · 3, but no periodic point of least period 2k−1 (2m+1) for any positive integer m. This is in contrast to the fact that there are expanding (but not piecewise monotonic) maps in C0(I, I) with very large expanding constants which have exactly one fixed point, say, at x = 1, but no other periodic point.

Lebesque measure zero subsets of the real line and an infinite game

1996 ◽  
Vol 61 (1) ◽  
pp. 246-249 ◽  
Author(s):  
Marion Scheepers
Keyword(s):  
Lebesgue Measure ◽  
Real Line ◽  
Measure Zero ◽  
Cardinal Number ◽  
Minimal Cardinality ◽  
Combinatorial Characterization ◽  
The Real ◽  
Lebesque Measure ◽  
The Ideal

Let denote the ideal of Lebesgue measure zero subsets of the real line. Then add() denotes the minimal cardinality of a subset of whose union is not an element of . In [1] Bartoszynski gave an elegant combinatorial characterization of add(), namely: add() is the least cardinal number κ for which the following assertion fails:For every family of at mostκ functions from ω to ω there is a function F from ω to the finite subsets of ω such that:1. For each m, F(m) has at most m + 1 elements, and2. for each f inthere are only finitely many m such that f(m) is not an element of F(m).The symbol A(κ) will denote the assertion above about κ. In the course of his proof, Bartoszynski also shows that the cardinality restriction in 1 is not sharp. Indeed, let (Rm: m < ω) be any sequence of integers such that for each m Rm, ≤ Rm+1, and such that limm→∞Rm = ∞. Then the truth of the assertion A(κ) is preserved if in 1 we say instead that1′. For each m, F(m) has at most Rm elements.We shall use this observation later on. We now define three more statements, denoted B(κ), C(κ) and D(κ), about cardinal number κ.

FIRST BETTI NUMBERS OF ABELIAN COVERINGS OF THE COMPLEX PROJECTIVE PLANE BRANCHED OVER LINE CONFIGURATIONS

2000 ◽  
Vol 09 (02) ◽  
pp. 271-284 ◽  
Author(s):  
IKUO TAYAMA
Keyword(s):  
Projective Plane ◽  
Real Line ◽  
General Position ◽  
Betti Numbers ◽  
Complex Projective Plane ◽  
The Real ◽  
Complex Projective

We prove the following results concerning the first Betti numbers of abelian coverings of CP2 branched over line configurations: (1) An estimate of the first Betti numbers. (2) A characterization of central and general position line configurations in terms of the first Betti numbers of abelian coverings. (3) The first Betti numbers of the abelian coverings of the real line configurations up to 7 components.

Subgradient Criteria for Monotonicity, The Lipschitz Condition, and Convexity

1993 ◽  
Vol 45 (6) ◽  
pp. 1167-1183 ◽  
Author(s):  
F. H. Clarke ◽  
R. J. Stern ◽  
P. R. Wolenski
Keyword(s):  
Hilbert Space ◽  
Lipschitz Condition ◽  
Real Line ◽  
Nonsmooth Analysis ◽  
Real Hilbert Space ◽  
Lower Semicontinuous ◽  
Multidimensional Case ◽  
The Real

AbstractLet ƒ H → (—∞,∞] be lower semicontinuous, where H is a real Hilbert space. An approach based upon nonsmooth analysis and optimization is used in order to characterize monotonicity of ƒ with respect to a cone, as well as Lipschitz behavior and constancy. The results, which involve hypotheses on the proximal subgradient ∂ πƒ, specialize on the real line to yield classical characterizations of these properties in terms of the Dini derivate. They also give new extensions of these results to the multidimensional case. A new proof of a known characterization of convexity in terms of proximal subgradient monotonicity is also given.

A characterization of order statistic point processes that are mixed Poisson processes and mixed sample processes simultaneously

1985 ◽  
Vol 22 (02) ◽  
pp. 314-323
Author(s):  
A. Deffner ◽  
E. Haeusler
Keyword(s):  
Order Statistic ◽  
Real Line ◽  
Time Scale ◽  
Point Processes ◽  
Present Note ◽  
Poisson Processes ◽  
Scale Transformation ◽  
The Real ◽  
Mixed Sample

The results of Nawrotzki (1962), Feigin (1979) and Puri (1982) show that the class of all point processes (on the real line) with the order statistic property consists of all mixed Poisson processes up to a time-scale transformation, and of all mixed sample processes. The present note characterizes those order statistic point processes that are mixed Poisson processes and mixed sample processes simultaneously.

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