Factorisation in modular group and pure periodic negative-regular continued fractions

The object of the talk is to present properties of pure periodic continued fractions of the form
\begin{equation*}\tag{1} \underbrace{ \cfrac{\!\!{}-1}{b_1}\underset{\displaystyle+}{}\, \cfrac{\!\!{}-1}{b_2}\underset{\displaystyle+\cdots+}{}\,\cfrac{\!\!{}-1}{b_n}}_n\underset{\displaystyle+}{}\, \underbrace{\cfrac{\!\!{}-1}{b_1}\underset{\displaystyle+}{}\, \cfrac{\!\!{}-1}{b_2}\underset{\displaystyle+\cdots+}{}\,\cfrac{\!\!{}-1}{b_n}}_n\underset{\displaystyle+\cdots{}}{}, \end{equation*}
where $b_k$ are (positive) integers. By Tietze's theorem, the fraction (1) converges to an irrational number if $|b_k|\geqslant2$ (except for the case $|b_k|=2$ for all $k$).
We have proved that without restriction $|b_k|\geqslant2$ the fraction (1) may converge to rational numbers or diverge. This is the deference between negative-regular continued fractions and classical regular continued fractions which always converge to irrational numbers.
Several algorithms for construction of periods $\{b_1,\ldots, b_n\}$ of periodic negative-regular continued fractions converging to rational numbers are given. The periods of a given length can be obtained by Fermat's infinite descent method applied to some Diophantine equations. An explicit simple formula for the minimal period for $x$ is presented. A construction using the Calkin-Wilf tree and Stern's diatomic series is described. Arbitrary primitive periods are in one-to-one correspondence with elements of the modular group $\Gamma$. Explicit formulas converting products of the standard generators $S$ and $ST$ in $\Gamma$ into primitive periods are obtained. The periods of elliptic elements of $\Gamma$ are completely described. This description results in a parametric formula for primitive periods of rational numbers. A pure periodic negative-regular continued fraction diverges if and only if either its period or its double or its triple represents the identity in $\Gamma$.
The talk is based on a joint work with Sergey Khrushchev (Satbayev University).