# Sh:741

- Göbel, R., & Shelah, S. (2002).
*Radicals and Plotkin’s problem concerning geometrically equivalent groups*. Proc. Amer. Math. Soc.,**130**(3), 673–674. arXiv: math/0010303 DOI: 10.1090/S0002-9939-01-06108-1 MR: 1866018 -
Abstract:

If G and X are groups and N is a normal subgroup of X, then the G-closure of N in X is the normal subgroup {\overline X}^G=\bigcap\{{\rm ker}\varphi|\varphi:X\rightarrow G, \text{ with } N \subseteq{\rm ker}\varphi\} of X. In particular, {\overline 1}^G = R_GX is the G-radical of X. Plotkin calls two groups G and H geometrically equivalent, written G\sim H, if for any free group F of finite rank and any normal subgroup N of F the G–closure and the H–closure of N in F are the same. Quasiidentities are formulas of the form (\bigwedge_{i\le n} w_i = 1 \rightarrow w =1) for any words w, w_i \ (i\le n) in a free group. Generally geometrically equivalent groups satisfy the same quasiidentiies. Plotkin showed that nilpotent groups G and H satisfy the same quasiidenties if and only if G and H are geometrically equivalent. Hence he conjectured that this might hold for any pair of groups. We provide a counterexample. - published version (2p)

Bib entry

@article{Sh:741, author = {G{\"o}bel, R{\"u}diger and Shelah, Saharon}, title = {{Radicals and Plotkin's problem concerning geometrically equivalent groups}}, journal = {Proc. Amer. Math. Soc.}, fjournal = {Proceedings of the American Mathematical Society}, volume = {130}, number = {3}, year = {2002}, pages = {673--674}, issn = {0002-9939}, mrnumber = {1866018}, mrclass = {20E06 (20E10)}, doi = {10.1090/S0002-9939-01-06108-1}, note = {\href{https://arxiv.org/abs/math/0010303}{arXiv: math/0010303}}, arxiv_number = {math/0010303} }