The Friedman embedding theorem
Copyright policy )The Friedman embedding theorem
The article is a sequel to the same author's article [in J. Group Theory 12, No. 4, 561-565 (2009; Zbl 1215.20034)] which already dealt with the ``Borelness'' of HNN constructions. In the former, the author showed that there exists \textit{no} Borel map \(\varphi\) from the space of countable groups to that of 2-generated groups such that: (1) any countable group \(G\) embeds into \(\varphi(G)\) and (2) if \(G\) and \(H\) are isomorphic, then so are \(\varphi(G)\) and \(\varphi(H)\). It was however claimed with no proof that there actually \textit{exists} a Borel map with properties (1) and (2) from the space of \textit{finitely generated} groups to that of 2-generated groups. The latter statement, attributed by the author to Harvey Friedman (for unpublished work), is the main theorem (Theorem 1.2) of the article under review. It is obtained by using some recursion theory, namely by first embedding a finitely-generated group \(G\) into the group of permutations of the integers with Turing degree not greater than the Turing degree of the word problem in \(G\). Another ingredient is a combinatorial group theoretic argument of \textit{F. Galvin} [Am. Math. Mon. 100, No. 6, 578-580 (1993; Zbl 0937.20501)]. In section 6 some conjectures are made, around the (non-)existence of a ``purely group-theoretic'' method, i.e. continuous \(\varphi\).
- Rutgers, The State University of New Jersey United States
- Rutgers University New Brunswick United States
Generators, relations, and presentations of groups, Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations, Recursion theory, Algebra and Number Theory, recursion theory, Subgroup theorems; subgroup growth, Turing degrees, Word problems, other decision problems, connections with logic and automata (group-theoretic aspects), Combinatorial group theory, combinatorial group theory, 2-generator groups, Higman-Neumann-Neumann extensions, Higman-Neumann-Neumann embedding theorem, Descriptive set theory, Word problems, etc. in computability and recursion theory, Borel equivalence relation, Borel equivalence relations, countable groups
Generators, relations, and presentations of groups, Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations, Recursion theory, Algebra and Number Theory, recursion theory, Subgroup theorems; subgroup growth, Turing degrees, Word problems, other decision problems, connections with logic and automata (group-theoretic aspects), Combinatorial group theory, combinatorial group theory, 2-generator groups, Higman-Neumann-Neumann extensions, Higman-Neumann-Neumann embedding theorem, Descriptive set theory, Word problems, etc. in computability and recursion theory, Borel equivalence relation, Borel equivalence relations, countable groups
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