The Andrews-Curtis conjecture states that every balanced presentation of the trivial group can be transformed into a trivial presentation by a sequence of Nielsen transformations on the relators together with conjugations of relators. It is named after James J. Andrews and Morton L. Curtis, who proposed it in 1965, and it is difficult to verify whether the conjecture holds for a given balanced presentation. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
StatementEvery balanced presentation of the trivial group can be transformed into a trivial presentation by a sequence of Nielsen transformations on the relators together with conjugations of relators. 1 Proposed Year Progress Toward ResolutionWidely believed false; no counterexamples are known, though there are numerous potential counterexamples. 1 Classification
Resolution Status Prize Status
Prize Status (category) Sources
1. Andrews-Curtis Conjecture (Wikipedia)
Wikimedia FoundationLead section
named after James J. Andrews and Morton L. Curtis who proposed it in 1965
Lead section, first sentence
every balanced presentation (i.e. a presentation with the same number of generators and relations) of the trivial group can be transformed into a trivial presentation by a sequence of Nielsen transformations on the relators together with conjugations of relators
Lead section, second paragraph
While there are no counterexamples known, there are numerous potential counterexamples.
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