Subsemigroups of virtually free groups: finite Malcev presentations and testing for freeness

• Published in: Mathematical proceedings of the Cambridge Philosophical Society Vol. 141; no. 1; pp. 57 - 66
• Main Authors: CAIN, ALAN J., ROBERTSON, EDMUND F., RUšKUC, NIK
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01.07.2006
• ISSN: 0305-0041; 1469-8064
• DOI: 10.1017/S0305004106009236

This paper shows that, given a finite subset $X$ of a finitely generated virtually free group $F$, the freeness of the subsemigroup of $F$ generated by $X$ can be tested algorithmically. (A group is virtually free if it contains a free subgroup of finite index.) It is then shown that every finitely generated subsemigroup of $F$ has a finite Malcev presentation (a type of semigroup presentation which can be used to define any semigroup that embeds in a group), and that such a presentation can be effectively found from any finite generating set.