Lehnert’s conjecture

For every finitely generated group $G$ , $G$ has co-context-free word problem if and only if $G$ embeds into Thompson's group $V$ . Explicitly, for a finite generating set $X$ of $G$ , the complement $(X\cup X^{-1})^*\setminus\operatorname{WP}(G,X)$ of the word problem is a context-free language if and only if there exists an injective homomorphism $G\hookrightarrow V$ .

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

Lehnert's conjecture via the universal group $QAut(T_{2,c})$
The conjecture is equivalently formulated by replacing Thompson's group $V$ with the universal group $QAut(T_{2,c})$ : a finitely generated group has co-context-free word problem if and only if it embeds into $QAut(T_{2,c})$ .
source: Background supplied with the source

Sources & referencesView supporting material

Primary source

arXiv

Additional references

arXiv:2608.02111v1 — arXiv

formal-languages group-theory

2 days ago

Progress summary

Refreshed 2 days ago

Partially solved

Recent work supports Lehnert’s conjecture and rules out several proposed counterexamples, but it does not prove the conjecture.

Lehnert’s conjecture asserts that a finitely generated group is co-context-free exactly when it embeds into Thompson’s group $V$ . Earlier work reformulated the conjecture using the universal group $QAut(T_{2,c})$ .

Known results

$QAut(T_{2,c})$ embeds into $V$ , giving an equivalent formulation of the conjecture (2013).

2026 embedding results

Two 2026 papers characterize embeddings into $V$ through finitely many context-free subgroups or faithful context-free actions. They unify all previously known co-context-free examples inside $V$ and exclude, among others, Grigorchuk’s group, the Basilica group, and the Hanoï Towers groups; an independent proof by Henry Jaspars is also reported. The authors explicitly present this as evidence, not a proof of Lehnert’s conjecture.

Current status (as of August 2026): Lehnert’s conjecture remains open; substantial embedding criteria and exclusions support it, but no proof or counterexample has been reported.

Sources

No solutions have been posted yet.

Equivalent formulations 1