Logic Seminar

Collin BleakUniversity of St. Andrews
Amending Lehnert's Conjecture

Friday, November 19, 2021 - 2:45pm
Malott 206

For a finitely generated group $G$ with generating set $X$, we say the word problem for $G$ (with respect to $X$) is the language of words (in generators and inverses) which equate to the identity in $G$. The coword problem of $G$ is then the complementary language. In a celebrated result of the 1980's, Muller and Schupp used Dunwoody Accessibility to show that the class CF of groups with context free word problem is precisely the class of finitely generated virtually free groups. In 2005, Holt, Rees, Roever and Thomas introduce the class CoCF of groups with context-free coword problem as a natural generalisation of the class of CF groups. In that initial paper, it was expected that R. Thompson's group $V$ was not in the class CoCF. However, Lehnert and Schweitzer show in 2007 that a group $\mathrm{QAut}(T_{2,c})$ is a CoCF group, and thus $V$, a subgroup of $\mathrm{QAut}(T_{2,c})$, is also CoCF. Lehnert also conjectured that a group is in CoCF if and only if it is finitely generated and embeds in $\mathrm{QAut}(T_{2,c})$. In this talk, we give an embedding of $\mathrm{QAut}(T_{2,c})$ into $V$, so that Lehnert's conjecture becomes: a group in the class CoCF if and only if it embeds as a finitely generated subgroup of $V$. Joint with Francesco Matucci.