Nielsen equivalence in Gupta-Sidki groups

Myropolska, Aglaia

doi:10.1090/proc/13612

Nielsen equivalence in Gupta-Sidki groups
HTML articles powered by AMS MathViewer

by Aglaia Myropolska PDF

Proc. Amer. Math. Soc. 145 (2017), 3331-3342 Request permission

Abstract:

For a group $G$ generated by $k$ elements, the Nielsen equivalence classes are defined as orbits of the action of $\textrm {Aut} F_k$, the automorphism group of the free group of rank $k$, on the set of generating $k$-tuples of $G$.

Let $p\geq 3$ be prime and $G_p$ the Gupta-Sidki $p$-group. We prove that there are infinitely many Nielsen equivalence classes on generating pairs of $G_p$.

References

J. J. Andrews and M. L. Curtis, Free groups and handlebodies, Proc. Amer. Math. Soc. 16 (1965), 192–195. MR 173241, DOI 10.1090/S0002-9939-1965-0173241-8
Theofanis Alexoudas, Benjamin Klopsch, and Anitha Thillaisundaram, Maximal subgroups of multi-edge spinal groups, Groups Geom. Dyn. 10 (2016), no. 2, 619–648. MR 3513111, DOI 10.4171/GGD/359
Persi Diaconis and Ronald Graham, The graph of generating sets of an abelian group, Colloq. Math. 80 (1999), no. 1, 31–38. MR 1684568, DOI 10.4064/cm-80-1-31-38
M. J. Dunwoody, On $T$-systems of groups, J. Austral. Math. Soc. 3 (1963), 172–179. MR 0153745
Martin J. Evans, Presentations of groups involving more generators than are necessary, Proc. London Math. Soc. (3) 67 (1993), no. 1, 106–126. MR 1218122, DOI 10.1112/plms/s3-67.1.106
M. J. Evans, Nielsen equivalence classes and stability graphs of finitely generated groups, Ischia group theory 2006, World Sci. Publ., Hackensack, NJ, 2007, pp. 103–119. MR 2405933, DOI 10.1142/9789812708670_{0}011
Wolfgang Gaschütz, Zu einem von B. H. und H. Neumann gestellten Problem, Math. Nachr. 14 (1955), 249–252 (1956) (German). MR 83993, DOI 10.1002/mana.19550140406
R. I. Grigorčuk, On Burnside’s problem on periodic groups, Funktsional. Anal. i Prilozhen. 14 (1980), no. 1, 53–54 (Russian). MR 565099
Narain Gupta and Saïd Sidki, On the Burnside problem for periodic groups, Math. Z. 182 (1983), no. 3, 385–388. MR 696534, DOI 10.1007/BF01179757
N. Gupta and Said Sidki, Some infinite $p$-groups, Algebra i Logika 22 (1983), no. 5, 584–589 (English, with Russian summary). MR 759409
Narain Gupta and Said Sidki, Extension of groups by tree automorphisms, Contributions to group theory, Contemp. Math., vol. 33, Amer. Math. Soc., Providence, RI, 1984, pp. 232–246. MR 767112, DOI 10.1090/conm/033/767112
Alexander Lubotzky, Dynamics of $\textrm {Aut}(F_N)$ actions on group presentations and representations, Geometry, rigidity, and group actions, Chicago Lectures in Math., Univ. Chicago Press, Chicago, IL, 2011, pp. 609–643. MR 2807845
A. Myropolska, The class $\mathcal {MN}$ of groups in which all maximal subgroups are normal, Preprint arXiv:1509.08090.
Aglaia Myropolska, Andrews-Curtis and Nielsen equivalence relations on some infinite groups, J. Group Theory 19 (2016), no. 1, 161–178. MR 3441132, DOI 10.1515/jgth-2015-0031
Volodymyr Nekrashevych, Self-similar groups, Mathematical Surveys and Monographs, vol. 117, American Mathematical Society, Providence, RI, 2005. MR 2162164, DOI 10.1090/surv/117
J. Nielsen, Über die Isomorphismen unendlicher Gruppen ohne Relation, Math. Ann. 79 (1918), no. 3, 269–272 (German). MR 1511927, DOI 10.1007/BF01458209
Bernhard H. Neumann and Hanna Neumann, Zwei Klassen charakteristischer Untergruppen und ihre Faktorgruppen, Math. Nachr. 4 (1951), 106–125 (German). MR 40297, DOI 10.1002/mana.19500040112
Daniel Oancea, A note on Nielsen equivalence in finitely generated abelian groups, Bull. Aust. Math. Soc. 84 (2011), no. 1, 127–136. MR 2817667, DOI 10.1017/S0004972711002279
Igor Pak, What do we know about the product replacement algorithm?, Groups and computation, III (Columbus, OH, 1999) Ohio State Univ. Math. Res. Inst. Publ., vol. 8, de Gruyter, Berlin, 2001, pp. 301–347. MR 1829489
E. L. Pervova, Everywhere dense subgroups of a group of tree automorphisms, Tr. Mat. Inst. Steklova 231 (2000), no. Din. Sist., Avtom. i Beskon. Gruppy, 356–367 (Russian, with Russian summary); English transl., Proc. Steklov Inst. Math. 4(231) (2000), 339–350. MR 1841763
E. L. Pervova, The congruence property of AT-groups, Algebra Logika 41 (2002), no. 5, 553–567, 634 (Russian, with Russian summary); English transl., Algebra Logic 41 (2002), no. 5, 306–313. MR 1953179, DOI 10.1023/A:1020979720331
E. L. Pervova, Maximal subgroups of some non locally finite $p$-groups, Internat. J. Algebra Comput. 15 (2005), no. 5-6, 1129–1150. MR 2197824, DOI 10.1142/S0218196705002803
Said Sidki, On a $2$-generated infinite $3$-group: the presentation problem, J. Algebra 110 (1987), no. 1, 13–23. MR 904179, DOI 10.1016/0021-8693(87)90034-2