Homomorphisms to acylindrically hyperbolic groups I: Equationally noetherian groups and families

Groves, D.; Hull, M.

doi:10.1090/tran/7789

Homomorphisms to acylindrically hyperbolic groups I: Equationally noetherian groups and families
HTML articles powered by AMS MathViewer

by D. Groves and M. Hull PDF

Trans. Amer. Math. Soc. 372 (2019), 7141-7190 Request permission

Abstract:

We study the set of homomorphisms from a fixed finitely generated group $G$ into a family of groups $\mathcal {G}$ which are ‘uniformly acylindrically hyperbolic’. Our main results reduce this study to sets of homomorphisms which do not diverge in an appropriate sense. As an application, we prove that any relatively hyperbolic group with equationally noetherian peripheral subgroups is itself equationally noetherian.

References

Emina Alibegović, Makanin-Razborov diagrams for limit groups, Geom. Topol. 11 (2007), 643–666. MR 2302499, DOI 10.2140/gt.2007.11.643
Gilbert Baumslag, Alexei Myasnikov, and Vladimir Remeslennikov, Algebraic geometry over groups. I. Algebraic sets and ideal theory, J. Algebra 219 (1999), no. 1, 16–79. MR 1707663, DOI 10.1006/jabr.1999.7881
Mladen Bestvina, Degenerations of the hyperbolic space, Duke Math. J. 56 (1988), no. 1, 143–161. MR 932860, DOI 10.1215/S0012-7094-88-05607-4
Mladen Bestvina and Mark Feighn, Stable actions of groups on real trees, Invent. Math. 121 (1995), no. 2, 287–321. MR 1346208, DOI 10.1007/BF01884300
Brian H. Bowditch, Tight geodesics in the curve complex, Invent. Math. 171 (2008), no. 2, 281–300. MR 2367021, DOI 10.1007/s00222-007-0081-y
Martin R. Bridson and André Haefliger, Metric spaces of non-positive curvature, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999. MR 1744486, DOI 10.1007/978-3-662-12494-9
M. R. Bridson and G. A. Swarup, On Hausdorff-Gromov convergence and a theorem of Paulin, Enseign. Math. (2) 40 (1994), no. 3-4, 267–289. MR 1309129
Montserrat Casals-Ruiz and Ilya V. Kazachkov, On systems of equations over free products of groups, J. Algebra 333 (2011), 368–426. MR 2785952, DOI 10.1016/j.jalgebra.2010.09.044
Christophe Champetier and Vincent Guirardel, Limit groups as limits of free groups, Israel J. Math. 146 (2005), 1–75. MR 2151593, DOI 10.1007/BF02773526
F. Dahmani, V. Guirardel, and D. Osin, Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces, Mem. Amer. Math. Soc. 245 (2017), no. 1156, v+152. MR 3589159, DOI 10.1090/memo/1156
B. Farb, Relatively hyperbolic groups, Geom. Funct. Anal. 8 (1998), no. 5, 810–840. MR 1650094, DOI 10.1007/s000390050075
M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75–263. MR 919829, DOI 10.1007/978-1-4613-9586-7_{3}
Daniel Groves, Limit groups for relatively hyperbolic groups. II. Makanin-Razborov diagrams, Geom. Topol. 9 (2005), 2319–2358. MR 2209374, DOI 10.2140/gt.2005.9.2319
Daniel Groves, Limit groups for relatively hyperbolic groups. I. The basic tools, Algebr. Geom. Topol. 9 (2009), no. 3, 1423–1466. MR 2530123, DOI 10.2140/agt.2009.9.1423
Daniel Groves and Henry Wilton, Enumerating limit groups, Groups Geom. Dyn. 3 (2009), no. 3, 389–399. MR 2516172, DOI 10.4171/GGD/63
V. S. Guba, Equivalence of infinite systems of equations in free groups and semigroups to finite subsystems, Mat. Zametki 40 (1986), no. 3, 321–324, 428 (Russian). MR 869922
Vincent Guirardel, Actions of finitely generated groups on $\Bbb R$-trees, Ann. Inst. Fourier (Grenoble) 58 (2008), no. 1, 159–211 (English, with English and French summaries). MR 2401220, DOI 10.5802/aif.2348
Vincent Guirardel and Gilbert Levitt, JSJ decompositions of groups, Astérisque 395 (2017), vii+165 (English, with English and French summaries). MR 3758992
Vincent Guirardel, Limit groups and groups acting freely on $\Bbb R^n$-trees, Geom. Topol. 8 (2004), 1427–1470. MR 2119301, DOI 10.2140/gt.2004.8.1427
Ch. K. Gupta and N. S. Romanovskiĭ, The property of being equational Noetherian of some solvable groups, Algebra Logika 46 (2007), no. 1, 46–59, 124 (Russian, with Russian summary); English transl., Algebra Logic 46 (2007), no. 1, 28–36. MR 2321079, DOI 10.1007/s10469-007-0003-5
G. Christopher Hruska and Bruce Kleiner, Hadamard spaces with isolated flats, Geom. Topol. 9 (2005), 1501–1538. With an appendix by the authors and Mohamad Hindawi. MR 2175151, DOI 10.2140/gt.2005.9.1501
HJRW (http://mathoverflow.net/users/1463/hjrw), Is there a non-hopfian lacunary hyperbolic group? MathOverflow, http://mathoverflow.net/q/75784 (version: 2011-09-28).
Eric Jaligot and Zlil Sela, Makanin-Razborov diagrams over free products, Illinois J. Math. 54 (2010), no. 1, 19–68. MR 2776984
O. Kharlampovich and A. Myasnikov, Irreducible affine varieties over a free group. I. Irreducibility of quadratic equations and Nullstellensatz, J. Algebra 200 (1998), no. 2, 472–516. MR 1610660, DOI 10.1006/jabr.1997.7183
O. Kharlampovich and 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), no. 2, 517–570. MR 1610664, DOI 10.1006/jabr.1997.7184
P. A. Linnell, On accessibility of groups, J. Pure Appl. Algebra 30 (1983), no. 1, 39–46. MR 716233, DOI 10.1016/0022-4049(83)90037-3
A. Yu. Ol′shanskiĭ, On residualing homomorphisms and $G$-subgroups of hyperbolic groups, Internat. J. Algebra Comput. 3 (1993), no. 4, 365–409. MR 1250244, DOI 10.1142/S0218196793000251
Denis V. Osin, Relatively hyperbolic groups: intrinsic geometry, algebraic properties, and algorithmic problems, Mem. Amer. Math. Soc. 179 (2006), no. 843, vi+100. MR 2182268, DOI 10.1090/memo/0843
D. Osin, Acylindrically hyperbolic groups, Trans. Amer. Math. Soc. 368 (2016), no. 2, 851–888. MR 3430352, DOI 10.1090/tran/6343
Abderezak Ould Houcine, Limit groups of equationally Noetherian groups, Geometric group theory, Trends Math., Birkhäuser, Basel, 2007, pp. 103–119. MR 2395792, DOI 10.1007/978-3-7643-8412-8_{8}
Frédéric Paulin, Outer automorphisms of hyperbolic groups and small actions on $\textbf {R}$-trees, Arboreal group theory (Berkeley, CA, 1988) Math. Sci. Res. Inst. Publ., vol. 19, Springer, New York, 1991, pp. 331–343. MR 1105339, DOI 10.1007/978-1-4612-3142-4_{1}2
C. Reinfeldt and R. Weidmann, Makanin-Razborov diagrams for hyperbolic groups, preprint, http://www.math.uni-kiel.de/algebra/de/weidmann/research/material-research/mr2014-pdf, 2014.
E. Rips and Z. Sela, Structure and rigidity in hyperbolic groups. I, Geom. Funct. Anal. 4 (1994), no. 3, 337–371. MR 1274119, DOI 10.1007/BF01896245
N. S. Romanovskiĭ, Equational Noetherianity of rigid solvable groups, Algebra Logika 48 (2009), no. 2, 258–279, 284, 287 (Russian, with English and Russian summaries); English transl., Algebra Logic 48 (2009), no. 2, 147–160. MR 2573021, DOI 10.1007/s10469-009-9045-1
Z. Sela, Diophantine geometry over groups: a list of research problems, http://www.ma.huji.ac.il/$\sim$zlil/problems.dvi.
Z. Sela, Acylindrical accessibility for groups, Invent. Math. 129 (1997), no. 3, 527–565. MR 1465334, DOI 10.1007/s002220050172
Zlil Sela, Diophantine geometry over groups. I. Makanin-Razborov diagrams, Publ. Math. Inst. Hautes Études Sci. 93 (2001), 31–105. MR 1863735, DOI 10.1007/s10240-001-8188-y
Z. Sela, Diophantine geometry over groups. VII. The elementary theory of a hyperbolic group, Proc. Lond. Math. Soc. (3) 99 (2009), no. 1, 217–273. MR 2520356, DOI 10.1112/plms/pdn052
Z. Sela, Diophantine geometry over groups. X: The elementary theory of free products, arXiv:1012.0044, 2010.
D. Vallino. Algebraic and definable closure in free groups, Thesis, Université Claude Bernard - Lyon I, June 2012.
L. van den Dries and A. J. Wilkie, Gromov’s theorem on groups of polynomial growth and elementary logic, J. Algebra 89 (1984), no. 2, 349–374. MR 751150, DOI 10.1016/0021-8693(84)90223-0
Richard Weidmann, On accessibility of finitely generated groups, Q. J. Math. 63 (2012), no. 1, 211–225. MR 2889188, DOI 10.1093/qmath/haq038
J. Wiegold, Groups with boundedly finite classes of conjugate elements, Proc. Roy. Soc. London Ser. A 238 (1957), 389–401. MR 83488, DOI 10.1098/rspa.1957.0007
Henry Wilton and Pavel Zalesskii, Profinite properties of graph manifolds, Geom. Dedicata 147 (2010), 29–45. MR 2660565, DOI 10.1007/s10711-009-9437-3