{"@context":{"@vocab":"https://cir.nii.ac.jp/schema/1.0/","rdfs":"http://www.w3.org/2000/01/rdf-schema#","dc":"http://purl.org/dc/elements/1.1/","dcterms":"http://purl.org/dc/terms/","foaf":"http://xmlns.com/foaf/0.1/","prism":"http://prismstandard.org/namespaces/basic/2.0/","cinii":"http://ci.nii.ac.jp/ns/1.0/","datacite":"https://schema.datacite.org/meta/kernel-4/","ndl":"http://ndl.go.jp/dcndl/terms/","jpcoar":"https://github.com/JPCOAR/schema/blob/master/2.0/"},"@id":"https://cir.nii.ac.jp/crid/1363670320119481856.json","@type":"Article","productIdentifier":[{"identifier":{"@type":"DOI","@value":"10.1090/s0002-9947-1991-1002920-5"}},{"identifier":{"@type":"URI","@value":"http://www.ams.org/tran/1991-326-01/S0002-9947-1991-1002920-5/S0002-9947-1991-1002920-5.pdf"}},{"identifier":{"@type":"URI","@value":"https://www.ams.org/tran/1991-326-01/S0002-9947-1991-1002920-5/S0002-9947-1991-1002920-5.pdf"}}],"dc:title":[{"@value":"The solution of length four equations over groups"}],"description":[{"type":"abstract","notation":[{"@value":"<p>Let <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\n  <mml:semantics>\n    <mml:mi>G</mml:mi>\n    <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\n  </mml:semantics>\n</mml:math>\n</inline-formula> be a group, <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper F\">\n  <mml:semantics>\n    <mml:mi>F</mml:mi>\n    <mml:annotation encoding=\"application/x-tex\">F</mml:annotation>\n  </mml:semantics>\n</mml:math>\n</inline-formula> the free group generated by <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"t\">\n  <mml:semantics>\n    <mml:mi>t</mml:mi>\n    <mml:annotation encoding=\"application/x-tex\">t</mml:annotation>\n  </mml:semantics>\n</mml:math>\n</inline-formula> and let <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"r left-parenthesis t right-parenthesis element-of upper G asterisk upper F\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi>r</mml:mi>\n      <mml:mo stretchy=\"false\">(</mml:mo>\n      <mml:mi>t</mml:mi>\n      <mml:mo stretchy=\"false\">)</mml:mo>\n      <mml:mo>∈</mml:mo>\n      <mml:mi>G</mml:mi>\n      <mml:mo>∗</mml:mo>\n      <mml:mi>F</mml:mi>\n    </mml:mrow>\n    <mml:annotation encoding=\"application/x-tex\">r(t) \\in G \\ast F</mml:annotation>\n  </mml:semantics>\n</mml:math>\n</inline-formula>. The equation <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"r left-parenthesis t right-parenthesis equals 1\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi>r</mml:mi>\n      <mml:mo stretchy=\"false\">(</mml:mo>\n      <mml:mi>t</mml:mi>\n      <mml:mo stretchy=\"false\">)</mml:mo>\n      <mml:mo>=</mml:mo>\n      <mml:mn>1</mml:mn>\n    </mml:mrow>\n    <mml:annotation encoding=\"application/x-tex\">r(t) = 1</mml:annotation>\n  </mml:semantics>\n</mml:math>\n</inline-formula> is said to have a solution over <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\n  <mml:semantics>\n    <mml:mi>G</mml:mi>\n    <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\n  </mml:semantics>\n</mml:math>\n</inline-formula> if it has a solution in some group that contains <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\n  <mml:semantics>\n    <mml:mi>G</mml:mi>\n    <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\n  </mml:semantics>\n</mml:math>\n</inline-formula>. This is equivalent to saying that the natural map <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G right-arrow mathematical left-angle upper G asterisk upper F vertical-bar r left-parenthesis t right-parenthesis mathematical right-angle\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi>G</mml:mi>\n      <mml:mo stretchy=\"false\">→</mml:mo>\n      <mml:mo fence=\"false\" stretchy=\"false\">⟨</mml:mo>\n      <mml:mi>G</mml:mi>\n      <mml:mo>∗</mml:mo>\n      <mml:mi>F</mml:mi>\n      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n        <mml:mo stretchy=\"false\">|</mml:mo>\n      </mml:mrow>\n      <mml:mi>r</mml:mi>\n      <mml:mo stretchy=\"false\">(</mml:mo>\n      <mml:mi>t</mml:mi>\n      <mml:mo stretchy=\"false\">)</mml:mo>\n      <mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo>\n    </mml:mrow>\n    <mml:annotation encoding=\"application/x-tex\">G \\to \\langle G \\ast F|r(t)\\rangle</mml:annotation>\n  </mml:semantics>\n</mml:math>\n</inline-formula> is injective. There is a conjecture (attributed to M. Kervaire and F. Laudenbach) that injectivity fails only if the exponent sum of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"t\">\n  <mml:semantics>\n    <mml:mi>t</mml:mi>\n    <mml:annotation encoding=\"application/x-tex\">t</mml:annotation>\n  </mml:semantics>\n</mml:math>\n</inline-formula> in <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"r left-parenthesis t right-parenthesis\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi>r</mml:mi>\n      <mml:mo stretchy=\"false\">(</mml:mo>\n      <mml:mi>t</mml:mi>\n      <mml:mo stretchy=\"false\">)</mml:mo>\n    </mml:mrow>\n    <mml:annotation encoding=\"application/x-tex\">r(t)</mml:annotation>\n  </mml:semantics>\n</mml:math>\n</inline-formula> is zero. In this paper we verify this conjecture in the case when the sum of the absolute values of the exponent of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"t\">\n  <mml:semantics>\n    <mml:mi>t</mml:mi>\n    <mml:annotation encoding=\"application/x-tex\">t</mml:annotation>\n  </mml:semantics>\n</mml:math>\n</inline-formula> in <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/M ..."}]}],"creator":[{"@id":"https://cir.nii.ac.jp/crid/1383670320119481857","@type":"Researcher","foaf:name":[{"@value":"Martin Edjvet"}]},{"@id":"https://cir.nii.ac.jp/crid/1383670320119481856","@type":"Researcher","foaf:name":[{"@value":"James Howie"}]}],"publication":{"publicationIdentifier":[{"@type":"PISSN","@value":"00029947"},{"@type":"EISSN","@value":"10886850"}],"prism:publicationName":[{"@value":"Transactions of the American Mathematical Society"}],"dc:publisher":[{"@value":"American Mathematical Society (AMS)"}],"prism:publicationDate":"1991","prism:volume":"326","prism:number":"1","prism:startingPage":"345","prism:endingPage":"369"},"reviewed":"false","url":[{"@id":"http://www.ams.org/tran/1991-326-01/S0002-9947-1991-1002920-5/S0002-9947-1991-1002920-5.pdf"},{"@id":"https://www.ams.org/tran/1991-326-01/S0002-9947-1991-1002920-5/S0002-9947-1991-1002920-5.pdf"}],"createdAt":"2012-06-06","modifiedAt":"2021-12-02","relatedProduct":[{"@id":"https://cir.nii.ac.jp/crid/1390001205229271680","@type":"Article","relationType":["isReferencedBy"],"jpcoar:relatedTitle":[{"@language":"en","@value":"ONE-RELATOR PRODUCTS OF TWO GROUPS OF ORDER THREE WITH SHORT RELATORS"}]}],"dataSourceIdentifier":[{"@type":"CROSSREF","@value":"10.1090/s0002-9947-1991-1002920-5"},{"@type":"CROSSREF","@value":"10.2206/kyushujm.52.81_references_DOI_FoUF9nDi7wTLjyuv4iEOwEBq7Tm"}]}