An exotic presentation of ℤ × ℤ and the Andrews–Curtis conjecture

Barmak, Jonathan Ariel¹

¹Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Buenos Aires, Argentina, Instituto de Investigaciones Matemáticas Luis A Santaló (IMAS), CONICET-Universidad de Buenos Aires, Buenos Aires, Argentina

Algebraic and Geometric Topology, Tome 25 (2025) no. 1, pp. 345-355

Résumé

We prove that the presentations 〈x,y∣[x,y],1〉 and 〈x,y∣[x,[x,y−1]]2y[y−1,x]y−1,[x,[[y−1,x],x]]〉 are not Q∗-equivalent even though their standard complexes have the same simple homotopy type.

Détail

DOI : 10.2140/agt.2025.25.345

Keywords: group presentations, $Q^*$-transformations, generalized Andrews–Curtis conjecture, simple homotopy type

Bibliographie

[1] J J Andrews, M L Curtis, Free groups and handlebodies, Proc. Amer. Math. Soc. 16 (1965) 192 | DOI

[2] S Bachmuth, H Y Mochizuki, E2≠SL2 for most Laurent polynomial rings, Amer. J. Math. 104 (1982) 1181 | DOI

[3] J A Barmak, The winding invariant, preprint (2019)

[4] H Bass, A Heller, R G Swan, The Whitehead group of a polynomial extension, Inst. Hautes Études Sci. Publ. Math. 22 (1964) 61 | DOI

[5] A V Borovik, A Lubotzky, A G Myasnikov, The finitary Andrews–Curtis conjecture, from: "Infinite groups: geometric, combinatorial and dynamical aspects" (editors L Bartholdi, T Ceccherini-Silberstein, T Smirnova-Nagnibeda, A Zuk), Progr. Math. 248, Birkhäuser (2005) 15 | DOI

[6] W J Browning, The effect of Curtis–Andrews moves on Jacobian matrices of perfect groups, unpublished manuscript (1976)

[7] W J Browning, Normal generators of finite groups, unpublished manuscript (1976)

[8] K Dennis, B Magurn, L Vaserstein, Generalized Euclidean group rings, J. Reine Angew. Math. 351 (1984) 113

[9] M J Evans, Primitive elements in the free metabelian group of rank 3, J. Algebra 220 (1999) 475 | DOI

[10] X Fernández, Morse theory for group presentations, Trans. Amer. Math. Soc. 377 (2024) 2495 | DOI

[11] J Harlander, A survey of recent progress on some problems in 2-dimensional topology, from: "Advances in two-dimensional homotopy and combinatorial group theory" (editors W Metzler, S Rosebrock), Lond. Math. Soc. Lect. Note Ser. 446, Cambridge Univ. Press (2017) 1 | DOI

[12] C Hog-Angeloni, W Metzler, The Andrews–Curtis conjecture and its generalizations, from: "Two-dimensional homotopy and combinatorial group theory" (editors C Hog-Angeloni, W Metzler, A J Sieradski), Lond. Math. Soc. Lect. Note Ser. 197, Cambridge Univ. Press (1993) 365 | DOI

[13] C Hog-Angeloni, W Metzler, Geometric aspects of two-dimensional complexes, from: "Two-dimensional homotopy and combinatorial group theory" (editors C Hog-Angeloni, W Metzler, A J Sieradski), Lond. Math. Soc. Lect. Note Ser. 197, Cambridge Univ. Press (1993) 1 | DOI

[14] C Hog-Angeloni, W Metzler, Further results concerning the Andrews–Curtis conjecture and its generalizations, from: "Advances in two-dimensional homotopy and combinatorial group theory" (editors W Metzler, S Rosebrock), Lond. Math. Soc. Lect. Note Ser. 446, Cambridge Univ. Press (2018) 27 | DOI

[15] M P Latiolais, Homotopy and homology classification of 2-complexes, from: "Two-dimensional homotopy and combinatorial group theory" (editors C Hog-Angeloni, W Metzler, A J Sieradski), Lond. Math. Soc. Lect. Note Ser. 197, Cambridge Univ. Press (1993) 97 | DOI

[16] M Lustig, Nielsen equivalence and simple-homotopy type, Proc. Lond. Math. Soc. 62 (1991) 537 | DOI

[17] W H Mannan, T Popiel, An exotic presentation of Q28, Algebr. Geom. Topol. 21 (2021) 2065 | DOI

[18] J Mccool, A Pietrowski, On a conjecture of W Magnus, from: "Word problems : decision problems and the Burnside problem in group theory", Stud. Logic Found. Math. 71, North-Holland (1973) 453 | DOI

[19] W Metzler, Über den Homotopietyp zweidimensionaler CW-Komplexe und Elementartransformationen bei Darstellungen von Gruppen durch Erzeugende und definierende Relationen, J. Reine Angew. Math. 285 (1976) 7 | DOI

[20] A D Myasnikov, A G Myasnikov, V Shpilrain, On the Andrews–Curtis equivalence, from: "Combinatorial and geometric group theory", Contemp. Math. 296, Amer. Math. Soc. (2002) 183 | DOI

[21] J Nicholson, Stably free modules and the unstable classification of 2-complexes, preprint (2021)

[22] A J Sieradski, Algebraic topology for two-dimensional complexes, from: "Two-dimensional homotopy and combinatorial group theory" (editors C Hog-Angeloni, W Metzler, A J Sieradski), Lond. Math. Soc. Lect. Note Ser. 197, Cambridge Univ. Press (1993) 51 | DOI

[23] R G Swan, Projective modules over Laurent polynomial rings, Trans. Amer. Math. Soc. 237 (1978) 111 | DOI

[24] H Zieschang, Über die Nielsensche Kürzungsmethode in freien Produkten mit Amalgam, Invent. Math. 10 (1970) 4 | DOI