Homomorphisms to 3-manifold groups

We prove foundational results about the set of homomorphisms from a finitely generated group to the collection of all fundamental groups of compact 3–manifolds and answer questions of Reid–Wang–Zhou [27] and Agol–Liu [3].

[1]  F. Paulin.  Topologie de Gromov équivariante, structures hyperboliques et arbres réels , 1988 .

[2]  Teruhiko Soma.  Epimorphism sequences between hyperbolic 3-manifold groups , 2001 .

[3]  G. Perelman.  Ricci flow with surgery on three-manifolds , 2003, math/0303109.

[4]  Qing Zhou,et al.  Generalized Hopfian Property, a Minimal Haken Manifold, and Epimorphisms Between 3-Manifold Groups , 2002 .

[5]  M. Bridson,et al.  Metric Spaces of Non-Positive Curvature , 1999 .

[6]  G. Levitt,et al.  Trees of cylinders and canonical splittings , 2008, 0811.2383.

[7]  PRESENTATION LENGTH AND SIMON’S CONJECTURE , 2011 .

[8]  Huai-Dong Cao,et al.  A Complete Proof of the Poincaré and Geometrization Conjectures - application of the Hamilton-Perelman theory of the Ricci flow , 2006 .

[9]  I. Agol,et al.  AN ALTERNATE PROOF OF WISE’S MALNORMAL SPECIAL QUOTIENT THEOREM , 2014, Forum of Mathematics, Pi.

[10]  B. H. Bowditch,et al.  Relatively hyperbolic Groups , 2012, Int. J. Algebra Comput..

[11]  Z. Sela,et al.  Diophantine geometry over groups I: Makanin-Razborov diagrams , 2001 .

[12]  Z. Sela.  Diophantine geometry over groups VII: The elementary theory of a hyperbolic group , 2009 .

[13]  R. Stephenson.  A and V , 1962, The British journal of ophthalmology.

[14]  HaiBaoDUAN.  Non-Zero Degree Maps Between 2n-Manifolds , 2004 .

[15]  G. Perelman.  The entropy formula for the Ricci flow and its geometric applications , 2002, math/0211159.

[16]  A. Myasnikov,et al.  Algebraic Geometry over Groups I. Algebraic Sets and Ideal Theory , 1999 .

[17]  Finiteness of nonzero degree maps between 3‐manifolds , 2011, Journal of Topology.

[18]  D. Groves,et al.  Recognizing geometric 3-manifold groups using the word problem , 2012, 1210.2101.

[19]  Gang Tian,et al.  The Geometrization Conjecture , 2014 .

[20]  Homomorphisms to acylindrically hyperbolic groups I: Equationally noetherian groups and families , 2017, Transactions of the American Mathematical Society.

[21]  V. Guirardel,et al.  Actions of finitely generated groups on R-trees , 2006, math/0607295.

[22]  P. Zalesskii,et al.  Profinite properties of graph manifolds , 2008, 0807.3727.

[23]  Alex Wilkie,et al.  Gromov's theorem on groups of polynomial growth and elementary logic , 1984 .

[24]  Peter A. Linnell.  On accessibility of groups , 1983 .

[25]  V. Guirardel,et al.  Jsj Decompositions of Groups , 2016, 1602.05139.

[26]  W. Thurston.  The geometry and topology of three-manifolds , 1979 .

[27]  W. Lueck,et al.  Topological rigidity for non-aspherical manifolds , 2005, math/0509238.

[28]  Bruce Kleiner,et al.  Notes on Perelman's papers , 2006 .

[29]  A. Myasnikov,et al.  Two Theorems about Equationally Noetherian Groups , 1997 .

[30]  Alexei Myasnikov,et al.  Algebraic Geometry over Groups , 2000 .

[31]  P. Scott,et al.  The geometries of 3-manifolds , 1983 .

[32]  R. Benedetti,et al.  Lectures on Hyperbolic Geometry , 1992 .

[33]  R. Weidmann,et al.  Makanin–Razborov diagrams for hyperbolic groups , 2020 .

[34]  Matthias Aschenbrenner,et al.  3-Manifold Groups , 2012, 1205.0202.

[35]  G. Perelman.  Finite extinction time for the solutions to the Ricci flow on certain three-manifolds , 2003, math/0307245.

[36]  Edited Rob Kirby,et al.  Problems in Low-Dimensional Topology , 1995 .