Residually free 3–manifolds

Wilton, Henry¹

¹Department of Mathematics, University of Texas, 1 University Station C1200, Austin, TX 78712-0257, USA

Algebraic and Geometric Topology, Tome 8 (2008) no. 4, pp. 2031-2047

Résumé

We classify those compact 3–manifolds with incompressible toral boundary whose fundamental groups are residually free. For example, if such a manifold M is prime and orientable and the fundamental group of M is nontrivial then M≅Σ × S1, where Σ is a surface.

Détail

DOI : 10.2140/agt.2008.8.2031

Keywords: 3-manifolds, free groups, geometric group theory

Bibliographie

[1] B Baumslag, Residually free groups, Proc. London Math. Soc. $(3)$ 17 (1967) 402

[2] J A Behrstock, W D Neumann, Quasi-isometric classification of graph manifold groups, Duke Math. J. 141 (2008) 217

[3] M Bestvina, M Feighn, Notes on Sela's work: Limit groups and Makanin–Razborov diagrams, to appear in “Geometry and cohomology in group theory (Durham, 2003)”

[4] M Bridson, J Howie, C F Iii, H Short, Subgroups of direct products of limit groups, to appear in Ann. of Math. $(2)$

[5] M R Bridson, H Wilton, Subgroup separability in residually free groups, Math. Z. 260 (2008) 25

[6] G Duchamp, D Krob, The lower central series of the free partially commutative group, Semigroup Forum 45 (1992) 385

[7] M Hall Jr., Subgroups of finite index in free groups, Canadian J. Math. 1 (1949) 187

[8] J Hempel, Residual finiteness for $3$–manifolds, from: "Combinatorial group theory and topology (Alta, Utah, 1984)", Ann. of Math. Stud. 111, Princeton Univ. Press (1987) 379

[9] W H Jaco, P B Shalen, Seifert fibered spaces in $3$–manifolds, Mem. Amer. Math. Soc. 21 (1979)

[10] K Johannson, Homotopy equivalences of $3$–manifolds with boundaries, Lecture Notes in Math. 761, Springer (1979)

[11] O Kharlampovich, A Myasnikov, Irreducible affine varieties over a free group. I. Irreducibility of quadratic equations and Nullstellensatz, J. Algebra 200 (1998) 472

[12] O Kharlampovich, A Myasnikov, Irreducible affine varieties over a free group. II. Systems in triangular quasi-quadratic form and description of residually free groups, J. Algebra 200 (1998) 517

[13] R C Lyndon, The equation $a^{2}b^{2}=c^{2}$ in free groups, Michigan Math. J 6 (1959) 89

[14] R C Lyndon, P E Schupp, Combinatorial group theory, Ergebnisse der Math. und ihrer Grenzgebiete 89, Springer (1977)

[15] W D Neumann, G A Swarup, Canonical decompositions of $3$–manifolds, Geom. Topol. 1 (1997) 21

[16] V N Remeslennikov, $\exists$–free groups, Sibirsk. Mat. Zh. 30 (1989) 193

[17] G P Scott, Finitely generated $3$–manifold groups are finitely presented, J. London Math. Soc. $(2)$ 6 (1973) 437

[18] P Scott, The geometries of $3$–manifolds, Bull. London Math. Soc. 15 (1983) 401

[19] Z Sela, Diophantine geometry over groups: a list of research problems, Preprint

[20] Z Sela, Diophantine geometry over groups. I. Makanin-Razborov diagrams, Publ. Math. Inst. Hautes Études Sci. (2001) 31