Algebraic & Geometric Topology Volume 25, issue 1 (2025)

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An exotic presentation of $\mathbb{Z} \times \mathbb{Z}$ and the Andrews–Curtis conjecture

Jonathan Ariel Barmak

Algebraic & Geometric Topology 25 (2025) 345–355

DOI: 10.2140/agt.2025.25.345

Bibliography

1	J J Andrews, M L Curtis, Free groups and handlebodies, Proc. Amer. Math. Soc. 16 (1965) 192 MR173241
2	S Bachmuth, H Y Mochizuki, E₂≠SL₂ for most Laurent polynomial rings, Amer. J. Math. 104 (1982) 1181 MR681732
3	J A Barmak, The winding invariant, preprint (2019) arXiv:1904.10072
4	H Bass, A Heller, R G Swan, The Whitehead group of a polynomial extension, Inst. Hautes Études Sci. Publ. Math. 22 (1964) 61 MR174605
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 MR2195451
6	W J Browning, The effect of Curtis–Andrews moves on Jacobian matrices of perfect groups, unpublished manuscript (1976)
7	W J Browning, Normal generators of finite groups, unpublished manuscript (1976)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 MR749679
9	M J Evans, Primitive elements in the free metabelian group of rank 3, J. Algebra 220 (1999) 475 MR1717353
10	X Fernández, Morse theory for group presentations, Trans. Amer. Math. Soc. 377 (2024) 2495 MR4744764
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 MR3752465
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 MR1279186
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 MR1279175
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 MR3752466
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 MR1279177
16	M Lustig, Nielsen equivalence and simple-homotopy type, Proc. Lond. Math. Soc. 62 (1991) 537 MR1095232
17	W H Mannan, T Popiel, An exotic presentation of Q₂₈, Algebr. Geom. Topol. 21 (2021) 2065 MR4302493
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 MR399269
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 MR440527
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 MR1921712
21	J Nicholson, Stably free modules and the unstable classification of 2-complexes, preprint (2021) arXiv:2108.02220
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 MR1279176
23	R G Swan, Projective modules over Laurent polynomial rings, Trans. Amer. Math. Soc. 237 (1978) 111 MR469906
24	H Zieschang, Über die Nielsensche Kürzungsmethode in freien Produkten mit Amalgam, Invent. Math. 10 (1970) 4 MR263929