Embeddings into simple free products

Meier, David

doi:10.1090/S0002-9939-1985-0773986-5

Embeddings into simple free products
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by David Meier

Proc. Amer. Math. Soc. 93 (1985), 387-392

DOI: https://doi.org/10.1090/S0002-9939-1985-0773986-5

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Abstract:

We prove that a countable group $G$ can be embedded into a two-generator simple group $S$ which is an amalgamated free product of groups $G * {F_1}$ and $F$, where $F$ and ${F_1}$ are free groups on two generators. $S$ is also the product of two commuting free subgroups. If $G$ has solvable word problem, then we can construct a recursive presentation for $S$.

References

Ruth Camm, Simple free products, J. London Math. Soc. 28 (1953), 66–76. MR 52420, DOI 10.1112/jlms/s1-28.1.66
William W. Boone and Graham Higman, An algebraic characterization of groups with soluble word problem, J. Austral. Math. Soc. 18 (1974), 41–53. Collection of articles dedicated to the memory of Hanna Neumann, IX. MR 0357625
A. P. Gorjuškin, Imbedding of countable groups in $2$-generator simple groups, Mat. Zametki 16 (1974), 231–235 (Russian). MR 382456
Roger C. Lyndon and Paul E. Schupp, Combinatorial group theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89, Springer-Verlag, Berlin-New York, 1977. MR 0577064

Combinatorial group theory

David Meier, A note on simple free products, Houston J. Math. 9 (1983), no. 4, 501–504. MR 732241
Paul E. Schupp, Embeddings into simple groups, J. London Math. Soc. (2) 13 (1976), no. 1, 90–94. MR 401932, DOI 10.1112/jlms/s2-13.1.90
Richard J. Thompson, Embeddings into finitely generated simple groups which preserve the word problem, Word problems, II (Conf. on Decision Problems in Algebra, Oxford, 1976), Studies in Logic and the Foundations of Mathematics, vol. 95, North-Holland, Amsterdam-New York, 1980, pp. 401–441. MR 579955