Proper actions and proper invariant metrics

We show that if a (locally compact) group $G$ acts properly on a locally compact $\sigma$-compact space $X$ then there is a family of $G$-invariant proper continuous finite-valued pseudometrics which induces the topology of $X$. If $X$ is furthermore metrizable then $G$ acts properly on $X$ if and only if there exists a $G$-invariant proper compatible metric on $X$.Comment: The paper has been completely rewritten and differs essentially from "Constructing invariant Heine-Borel metrics for proper G-spaces". The main result extended to the more general case when $G$ is a topological group which acts properly on a locally compact $\sigma$-compact Hausdorff space $X$. Note that there is a gap in the proof of Theorem 2.4 of the old versio

Combing Euclidean buildings

For an arbitrary Euclidean building we define a certain combing, which satisfies the `fellow traveller property' and admits a recursive definition. Using this combing we prove that any group acting freely, cocompactly and by order preserving automorphisms on a Euclidean building of one of the types A_n,B_n,C_n admits a biautomatic structure.Comment: 32 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTVol4/paper2.abs.htm

Combing Euclidean buildings

For an arbitrary Euclidean building we define a certain combing, which satisfies the "fellow traveller property" and admits a recursive definition. Using this combing we prove that any group acting freely, cocompactly and by order preserving automorphisms on a Euclidean building of one of the types A n ; B n ; C n admits a biautomatic structure. Contents 1 Introduction 3 2 Euclidean Coxeter complexes 4 3 Ordering Euclidean buildings 10 4 Definition of a combing 16 5 Fellow traveller property 22 6 Recursiveness of a combing C 25 7 Automatic structure for groups acting on Euclidean buildings of type A n ; B n ; C n 31 8 Appendix: Geodesicity of local geodesics 36 1 Introduction Let G be a group that acts properly and cocompactly on a piecewise Euclidean simply connected CAT (0)-complex \Delta (see e.g. [Bri91] for definitions). (The action of course is supposed to be cellular, properness means that the isotropy group G oe is finite for every cell oe and cocompactness means that \D..

The Topological Generating Rank of Solvable Lie Groups

Abels H, Noskov GA. The Topological Generating Rank of Solvable Lie Groups. JOURNAL OF LIE THEORY. 2019;29(2):457-471.We define the topological generating rank d (G) of a connected Lie group G as the minimal number of elements of G needed to generate a dense subgroup of G. We answer the following question posed by K. H. Hofmann and S.A. Morris [see: Finitely generated connected locally compact groups, J. Lie Theory (formerly Sem. Sophus Lie) 2(2) (1992) 123-134]: What is the topological generating rank of a connected solvable Lie group? If G is solvable we can reduce the question to the case that G is metabelian. We can furthermore reduce to the case that the natural representation of Q:= G(ab):= G/(G') over bar on A := (G') over bar is semisimple. Then d (G) is the maximum of the following two numbers: d (Q) and one plus the maximum of the multiplicities of the non-trivial isotypic components of the RQ-module A

Current Trends of Development of the Regional Systems of Earth Remote Sensing

В статье обсуждаются вопросы построения региональной многоцелевой системы дистанционного мониторинга Земли как системы решения задач. Сформулирован набор факторов, влияющих на создание и функционирование систем этого класса, рассмотрены основные функциональные блокиThe article discusses the construction of a regional multi-purpose remote monitoring of the Earth system as a problem-solving system. Formulate a set of factors that influence the establishment and operation of this class of systems , the basic function block

The Space and Terrestrial Weather Variations as Possible Factors for Ischemia Events in Saint Petersburg

The Space and Terrestrial Weather (Weather Complex) impact on ischemia cases in Saint Petersburg is investigated. The results show the main feature of the Weather Complex when it was related to the days of the different ischemia situations in the different ischemia people gender groups. The data treatment was done with some elements of the Folder Epochs Method, Cluster Analysis and the Mann–Whitney hypothesis test criterion