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.

All articles are
published in Russian.

Location: http://mech.math.msu.su/~fpm/eng/k00/k003/k00309h.htm

Last modified: December 8, 2000