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Register a new userGroebner-Shirshov bases for some one-relator groups
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In this paper, we prove that two-generator one-relator groups with depth less than or equal to 3 can be effectively embedded into a tower of HNN-extensions in which each group has the effective standard normal form. We give an example to show how to deal with some general cases for one-relator groups. By using the Magnus method and Composition-Diamond Lemma, we reprove the G. Higman, B. H. Neumann and H. Neumanns embedding theorem.
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A new construction of a free inverse semigroup was obtained by Poliakova and Schein in 2005. Based on their result, we find a Groebner-Shirshov basis of a free inverse semigroup relative to the deg-lex order of words. In particular, we give the (unique and shortest) Groebner-Shirshov normal forms in the classes of equivalent words of a free inverse semigroup together with the Groebner-Shirshov algorithm to transform any word to its normal form.
In this survey article, we report some new results of Groebner-Shirshov bases, including new Composition-Diamond lemmas, applications of some known Composition-Diamond lemmas and content of some expository papers.
In this paper, we define the Grobner-Shirshov basis for a dialgebra. The Composition-Diamond lemma for dialgebras is given then. As results, we give Grobner-Shirshov bases for the universal enveloping algebra of a Leibniz algebra, the bar extension of a dialgebra, the free product of two dialgebras, and Clifford dialgebra. We obtain some normal forms for algebras mentioned the above.
We prove a freeness theorem for low-rank subgroups of one-relator groups. Let $F$ be a free group, and let $win F$ be a non-primitive element. The primitivity rank of $w$, $pi(w)$, is the smallest rank of a subgroup of $F$ containing $w$ as an imprimitive element. Then any subgroup of the one-relator group $G=F/langlelangle wranglerangle$ generated by fewer than $pi(w)$ elements is free. In particular, if $pi(w)>2$ then $G$ doesnt contain any Baumslag--Solitar groups. The hypothesis that $pi(w)>2$ implies that the presentation complex $X$ of the one-relator group $G$ has negative immersions: if a compact, connected complex $Y$ immerses into $X$ and $chi(Y)geq 0$ then $Y$ is Nielsen equivalent to a graph. The freeness theorem is a consequence of a dependence theorem for free groups, which implies several classical facts about free and one-relator groups, including Magnus Freiheitssatz and theorems of Lyndon, Baumslag, Stallings and Duncan--Howie. The dependence theorem strengthens Wises $w$-cycles conjecture, proved independently by the authors and Helfer--Wise, which implies that the one-relator complex $X$ has non-positive immersions when $pi(w)>1$.
In this paper, we review Shirshovs method for free Lie algebras invented by him in 1962 which is now called the Groebner-Shirshov bases theory.
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