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Some quadratic equations in the free group of rank 2

Daciberg Gonçalves, Elena Kudryavtseva and Heiner Zieschang

Geometry & Topology Monographs 14 (2008) 219–294

DOI: 10.2140/gtm.2008.14.219

Bibliography
1 S Bogatyi, D L Gonçalves, E Kudryavtseva, H Zieschang, On the Wecken property for the root problem of mappings between surfaces, Mosc. Math. J. 3 (2003) 1223–1245 MR2058797
2 S Bogatyi, D L Gonçalves, H Zieschang, The minimal number of roots of surface mappings and quadratic equations in free groups, Math. Z. 236 (2001) 419–452 MR1821299
3 S A Bogatyĭ, D L Gonçalves, H Zieschang, Coincidence theory: the minimization problem, Tr. Mat. Inst. Steklova 225 (1999) 52–86 MR1725933
4 R Brooks, Certain subgroups of the fundamental group and the number of roots of f(x)=a, Amer. J. Math. 95 (1973) 720–728 MR0346777
5 K S Brown, Cohomology of groups, Graduate Texts in Mathematics 87, Springer (1994) MR1324339 Corrected reprint of the 1982 original
6 R F Brown, H Schirmer, Nielsen root theory and Hopf degree theory, Pacific J. Math. 198 (2001) 49–80 MR1831972
7 D E Cohen, R C Lyndon, Free bases for normal subgroups of free groups, Trans. Amer. Math. Soc. 108 (1963) 526–537 MR0170930
8 M Culler, Using surfaces to solve equations in free groups, Topology 20 (1981) 133–145 MR605653
9 D B A Epstein, The degree of a map, Proc. London Math. Soc. (3) 16 (1966) 369–383 MR0192475
10 D L Gonçalves, Coincidence of maps between surfaces, J. Korean Math. Soc. 36 (1999) 243–256 MR1688777
11 D L Gonçalves, E Kudryavtseva, H Zieschang, Intersection index of curves on surfaces and applications to quadratic equations in free groups, Atti Sem. Mat. Fis. Univ. Modena 49 (2001) 339–400 MR1881102
12 D L Gonçalves, E Kudryavtseva, H Zieschang, Roots of mappings on nonorientable surfaces and equations in free groups, Manuscripta Math. 107 (2002) 311–341 MR1906200
13 D L Gonçalves, H Zieschang, Equations in free groups and coincidence of mappings on surfaces, Math. Z. 237 (2001) 1–29 MR1836771
14 R I Grigorchuk, P F Kurchanov, On quadratic equations in free groups, from: "Proceedings of the International Conference on Algebra, Part 1 (Novosibirsk, 1989)", Contemp. Math. 131, Amer. Math. Soc. (1992) 159–171 MR1175769
15 R I Grigorchuk, P F Kurchanov, H Zieschang, Equivalence of homomorphisms of surface groups to free groups and some properties of 3–dimensional handlebodies, from: "Proceedings of the International Conference on Algebra, Part 1 (Novosibirsk, 1989)", Contemp. Math. 131, Amer. Math. Soc. (1992) 521–530 MR1175803
16 P Hilton, Nilpotent actions on nilpotent groups, from: "Algebraic and logic (Fourteenth Summer Res. Inst., Austral. Math. Soc., Monash Univ., Clayton, 1974)", Lecture Notes in Mathematics 450, Springer (1975) 174–196 MR0382447
17 P Hilton, G Mislin, J Roitberg, Localization of nilpotent groups and spaces, North-Holland Mathematics Studies 15, North-Holland Publishing Co. (1975) MR0478146
18 J I Hmelevskiĭ, Systems of equations in a free group I, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971) 1237–1268 MR0313395
19 J I Hmelevskiĭ, Systems of equations in a free group II, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972) 110–179 MR0313395
20 J I Hmelevskiĭ, Equations in free semigroups, Amer. Math. Soc. (1976) MR0393284 Translated by G A Kandall from the Russian original: Trudy Mat. Inst. Steklov. 107 (1971)
21 G Hochschild, J P Serre, Cohomology of group extensions, Trans. Amer. Math. Soc. 74 (1953) 110–134 MR0052438
22 H Hopf, Zur Topologie der Abbildungen von Mannigfaltigkeiten, Math. Ann. 102 (1930) 562–623 MR1512596
23 H Kneser, Glättung von Flächenabbildungen, Math. Ann. 100 (1928) 609–617 MR1512504
24 H Kneser, Die kleinste Bedeckungszahl innerhalb einer Klasse von Flächenabbildungen, Math. Ann. 103 (1930) 347–358 MR1512626
25 P H Kropholler, P A Linnell, J A Moody, Applications of a new K-theoretic theorem to soluble group rings, Proc. Amer. Math. Soc. 104 (1988) 675–684 MR964842
26 E Kudryavtseva, R Weidmann, H Zieschang, Quadratic equations in free groups and topological applications, from: "Recent advances in group theory and low-dimensional topology (Pusan, 2000)", Res. Exp. Math. 27, Heldermann (2003) 83–122 MR2004634
27 R C Lyndon, Cohomology theory of groups with a single defining relation, Ann. of Math. (2) 52 (1950) 650–665 MR0047046
28 R C Lyndon, The equation a2b2=c2 in free groups, Michigan Math. J 6 (1959) 89–95 MR0103218
29 R C Lyndon, Equations in free groups, Trans. Amer. Math. Soc. 96 (1960) 445–457 MR0151503
30 R C Lyndon, P E Schupp, Combinatorial group theory, Ergebnisse der Mathematik und ihrer Grenzgebiete 89, Springer (1977) MR0577064
31 W Magnus, A Karrass, D Solitar, Combinatorial group theory, Dover Publications Inc. (2004) MR2109550 Reprint of the 1976 second edition
32 G S Makanin, Equations in a free group, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982) 1199–1273, 1344 MR682490
33 J Nielsen, Untersuchungen zur Topologie der geschlossenen zweiseitigen Flächen, Acta Math. 50 (1927) 189–358 MR1555256
34 A Y Ol'shanskiĭ, Diagrams of homomorphisms of surface groups, Sibirsk. Mat. Zh. 30 (1989) 150–171 MR1043443
35 P Olum, Mappings of manifolds and the notion of degree, Ann. of Math. (2) 58 (1953) 458–480 MR0058212
36 R P Osborne, H Zieschang, Primitives in the free group on two generators, Invent. Math. 63 (1981) 17–24 MR608526
37 A A Razborov, Systems of equations in a free group, Izv. Akad. Nauk SSSR Ser. Mat. 48 (1984) 779–832 MR755958
38 O Schreier, Die Untergruppen der freien Gruppen, Abh. Math. Sem. Univ. Hamburg 5 (1927) 161–183
39 R Skora, The degree of a map between surfaces, Math. Ann. 276 (1987) 415–423 MR875337
40 J Stallings, Homology and central series of groups, J. Algebra 2 (1965) 170–181 MR0175956
41 A Steinberg, On equations in free groups, Michigan Math. J. 18 (1971) 87–95 MR0289614
42 A A Vdovina, Constructing of orientable Wicks forms and estimation of their number, Comm. Algebra 23 (1995) 3205–3222 MR1335298
43 A Vdovina, On the number of nonorientable Wicks forms in a free group, Proc. Roy. Soc. Edinburgh Sect. A 126 (1996) 113–116 MR1378835
44 A Vdovina, Products of commutators in free products, Internat. J. Algebra Comput. 7 (1997) 471–485 MR1459623
45 M J Wicks, The equation X²Y²=g over free products, from: "Proc. 2nd Congress Singapore Nat. Acad. Sci. 1971, Sci. Urban Environment Tropics" (1973) 238–248
46 N J Wicks, Commutators in free products, J. London Math. Soc. 37 (1962) 433–444 MR0142610
47 H Zieschang, Alternierende Produkte in freien Gruppen, Abh. Math. Sem. Univ. Hamburg 27 (1964) 13–31 MR0161901
48 H Zieschang, Alternierende Produkte in freien Gruppen II, Abh. Math. Sem. Univ. Hamburg 28 (1965) 219–233
49 H Zieschang, Discrete groups of plane motions and plane group images, Uspehi Mat. Nauk 21 (1966) 195–212 MR0195954
50 H Zieschang, E Vogt, H D Coldewey, Surfaces and planar discontinuous groups, Lecture Notes in Mathematics 835, Springer (1980) MR606743 Translated from the German by John Stillwell