Geodesic
Some relations between the word and divisibility problems in groups and semigroups
Izvestiya. Mathematics , Tome 15 (1980) no. 1, pp. 161-171.

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This paper studies the relationship between the word problems in a finitely presented semigroup $\Pi$, which is embeddable in a group, and in the group $\Gamma$ with the same generators and defining relations. We construct an example showing that even in the case when not only the word problem but also the left and right divisibility problems are solvable in $\Pi$, the word problem in $\Gamma$ may be unsolvable. Furthermore, we prove that the additional condition of the absence of cycles in the system of defining relations of $\Pi$ issufficient for the solvability of its word and divisibility problems to imply the solvability of the word problem in $\Gamma$. Bibliography: 3 titles. 
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O. A. Sarkisyan. Some relations between the word and divisibility problems in groups and semigroups. Izvestiya. Mathematics , Tome 15 (1980) no. 1, pp. 161-171. http://geodesic.mathdoc.fr/item/IM2_1980_15_1_a6/

[1] Adyan S. I., Opredelyayuschie sootnosheniya i algoritmicheskie problemy dlya grupp i polugrupp, Tr. Matem. in-ta im. V. A. Steklova AN SSSR, 85, 1966 | Zbl

[2] Borisov V. V., “Prostye primery grupp s nerazreshimoi problemoi tozhdestva”, Matem. zametki, 6:5 (1969), 521–532 | MR | Zbl

[3] Sarkisyan O. A., “O svyazi mezhdu algoritmicheskimi problemami v gruppakh i polugruppakh”, Dokl. AN SSSR, 227:6 (1976), 1305–1307 | MR | Zbl