Mathematics and Statistics Vol. 8(2), pp. 121 - 125
DOI: 10.13189/ms.2020.080207
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Geometric Topics on Elementary Amenable Groups

Mostafa Ftouhi ^*, Mohammed Barmaki , Driss Gretete
Ecole nationale des sciences appliquées Université Ibn Tofail campus universitaire, Morocco

ABSTRACT

The class of amenable groups plays an important role in many areas of mathematics such as ergodic theory, harmonic analysis, representation theory, dynamical systems, geometric group theory, probability theory and statistics. The class of amenable groups contains in particular all finite groups, all abelian groups and, more generally, all solvable groups. It is closed under the operations of taking subgroups, taking quotients, taking extensions, and taking inductive limits. In 1959, Harry Kesten proved that there is a relation between the amenability and the estimates of symmetric random walk on finitely generated groups. In this article we study the classification of locally compact compactly generated groups according to return probability to the origin. Our aim is to compare several geometric classes of groups. The central tool in this comparison is the return probability on locally compact groups. we introduce several classes of groups in order to characterize the geometry of locally compact groups compactly generated. Our aim is to compare these classes in order to better understand the geometry of such groups by referring to the behavior of random walks on these groups. As results, we have found inclusion relationships between these defined classes and we have given counterexamples for reciprocal inclusions.

KEYWORDS
Graphs and Groups; Subgroup Growth;Wreath Product; Probability Measures on Groups; Geometric Probability, Random Walks

Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] Mostafa Ftouhi , Mohammed Barmaki , Driss Gretete , "Geometric Topics on Elementary Amenable Groups," Mathematics and Statistics, Vol. 8, No. 2, pp. 121 - 125, 2020. DOI: 10.13189/ms.2020.080207.

(b). APA Format:
Mostafa Ftouhi , Mohammed Barmaki , Driss Gretete (2020). Geometric Topics on Elementary Amenable Groups. Mathematics and Statistics, 8(2), 121 - 125. DOI: 10.13189/ms.2020.080207.