FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2000, VOLUME 6, NUMBER 3, PAGES 757-776

Roots in the universal covering group of the unimodular $2$´ 2-matrix group

T. V. Dubrovina
N. I. Dubrovin

Abstract

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The equation $xn=g$ has been solved in the universal covering group $$G of the group $SL(2)$. If $g$ is not a central element, then the $n$-th root of $g$ exists and is unique. In the case when $g$ belongs to the center of the universal covering $$G, the set of all solutions may be empty or may form a two-dimensional submanifold of the manifold $$G. The following two questions are considered. (A) How wide may be this submanifold from the algebraic point of view? (B) How can we complete the group $$G with absent roots?

Of the results close to the main theorem one can mention the following: the semigroup $SL(2)+$, consisting of all matrices $A$Î SL(2) with non-negative coefficients, is complete, that is one can derive any root from any element.

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Last modified: December 8, 2000