Roots in the universal covering group of the unimodular $2\times2$-matrix group

T. V. Dubrovina ; N. I. Dubrovin

Fundamentalʹnaâ i prikladnaâ matematika, Tome 6 (2000) no. 3, pp. 757-776.

Résumé

The equation $x^n=g$ has been solved in the universal covering group $\mathbb G$ of the group $\mathop{\mathrm{SL}}(2)$. 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 $\mathbb G$, the set of all solutions may be empty or may form a two-dimensional submanifold of the manifold $\mathbb G$. 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 $\mathbb G$ with absent roots? Of the results close to the main theorem one can mention the following: the semigroup $\mathop{\mathrm{SL}}(2)^+$, consisting of all matrices $A\in\mathop{\mathrm{SL}}(2)$ with non-negative coefficients, is complete, that is one can derive any root from any element.