Virtual retraction and Howson’s theorem in pro-𝑝 groups

Shusterman, Mark; Zalesskii, Pavel

doi:10.1090/tran/7784

Virtual retraction and Howson’s theorem in pro-$p$ groups
HTML articles powered by AMS MathViewer

by Mark Shusterman and Pavel Zalesskii PDF

Trans. Amer. Math. Soc. 373 (2020), 1501-1527 Request permission

Abstract:

We show that for every finitely generated closed subgroup $K$ of a nonsolvable Demushkin group $G$, there exists an open subgroup $U$ of $G$ containing $K$ and a continuous homomorphism $\tau \colon U \to K$ satisfying $\tau (k) = k$ for every $k \in K$. We prove that the intersection of a pair of finitely generated closed subgroups of a Demushkin group is finitely generated (giving an explicit bound on the number of generators). Furthermore, we show that these properties of Demushkin groups are preserved under free pro-$p$ products and deduce that Howson’s theorem holds for the Sylow subgroups of the absolute Galois group of a number field. Finally, we confirm two conjectures of Ribes, thus classifying the finitely generated pro-$p$ M. Hall groups.

References

G. N. Arzhantseva, A property of subgroups of infinite index in a free group, Proc. Amer. Math. Soc. 128 (2000), no. 11, 3205–3210. MR 1694447, DOI 10.1090/S0002-9939-00-05508-8
Karl Auinger and Benjamin Steinberg, Varieties of finite supersolvable groups with the M. Hall property, Math. Ann. 335 (2006), no. 4, 853–877. MR 2232019, DOI 10.1007/s00208-006-0767-2
Lior Bary-Soroker, Moshe Jarden, and Danny Neftin, The Sylow subgroups of the absolute Galois group $\textrm {Gal}(\Bbb {Q})$, Adv. Math. 284 (2015), 186–212. MR 3391075, DOI 10.1016/j.aim.2015.05.017
Benjamin Baumslag, Intersections of finitely generated subgroups in free products, J. London Math. Soc. 41 (1966), 673–679. MR 199247, DOI 10.1112/jlms/s1-41.1.673
Dave Benson, Nicole Lemire, Ján Mináč, and John Swallow, Detecting pro-$p$-groups that are not absolute Galois groups, J. Reine Angew. Math. 613 (2007), 175–191. MR 2377134, DOI 10.1515/CRELLE.2007.096
Nicolas Bergeron, Frédéric Haglund, and Daniel T. Wise, Hyperplane sections in arithmetic hyperbolic manifolds, J. Lond. Math. Soc. (2) 83 (2011), no. 2, 431–448. MR 2776645, DOI 10.1112/jlms/jdq082
O. V. Bogopol’ski, Finitely generated groups with M. Hall property, Algebra and Logic, 31 (1992), no. 3, 141–169.
O. V. Bogopol’skii, Almost free groups and the M. Hall property, Algebra and Logic, 33 (1994), no. 1, 1–13.
A. M. Brunner and R. G. Burns, Groups in which every finitely generated subgroup is almost a free factor, Canadian J. Math. 31 (1979), no. 6, 1329–1338. MR 553165, DOI 10.4153/CJM-1979-110-2
R. G. Burns, On finitely generated subgroups of free products, J. Austral. Math. Soc. 12 (1971), 358–364. MR 0311778
Robert G. Burns, On the intersection of finitely generated subgroups of a free group, Math. Z. 119 (1971), 121–130. MR 279166, DOI 10.1007/BF01109964
R. G. Burns, On the finitely generated subgroups of an amalgamated product of two groups, Trans. Amer. Math. Soc. 169 (1972), 293–306. MR 372043, DOI 10.1090/S0002-9947-1972-0372043-5
R. G. Burns, Finitely generated subgroups of $\textbf {HNN}$ groups, Canadian J. Math. 25 (1973), 1103–1112. MR 330305, DOI 10.4153/CJM-1973-117-7
R. G. Burns, On the rank of the intersection of subgroups of a Fuchsian group, Proceedings of the Second International Conference on the Theory of Groups (Australian Nat. Univ., Canberra, 1973) Lecture Notes in Math., Vol. 372, Springer, Berlin, 1974, pp. 165–187. MR 0367082
Yves de Cornulier, Finitely presented wreath products and double coset decompositions, Geom. Dedicata 122 (2006), 89–108. MR 2295543, DOI 10.1007/s10711-006-9061-4
François Dahmani, Combination of convergence groups, Geom. Topol. 7 (2003), 933–963. MR 2026551, DOI 10.2140/gt.2003.7.933
Warren Dicks, Equivalence of the strengthened Hanna Neumann conjecture and the amalgamated graph conjecture, Invent. Math. 117 (1994), no. 3, 373–389. MR 1283723, DOI 10.1007/BF01232249
Warren Dicks and Edward Formanek, The rank three case of the Hanna Neumann conjecture, J. Group Theory 4 (2001), no. 2, 113–151. MR 1812321, DOI 10.1515/jgth.2001.012
Warren Dicks and S. V. Ivanov, On the intersection of free subgroups in free products of groups, Math. Proc. Cambridge Philos. Soc. 144 (2008), no. 3, 511–534. MR 2418703, DOI 10.1017/S0305004107001041
J. D. Dixon, M. P. F. du Sautoy, A. Mann, and D. Segal, Analytic pro-$p$ groups, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 61, Cambridge University Press, Cambridge, 1999. MR 1720368, DOI 10.1017/CBO9780511470882
D. Dummit and J. Labute, On a new characterization of Demuskin groups, Invent. Math. 73 (1983), no. 3, 413–418. MR 718938, DOI 10.1007/BF01388436
Ido Efrat, Finitely generated pro-$p$ absolute Galois groups over global fields, J. Number Theory 77 (1999), no. 1, 83–96. MR 1695702, DOI 10.1006/jnth.1999.2379
Ido Efrat, Demuškin fields with valuations, Math. Z. 243 (2003), no. 2, 333–353. MR 1961869, DOI 10.1007/s00209-002-0471-1
Brent Everitt, Graphs, free groups and the Hanna Neumann conjecture, J. Group Theory 11 (2008), no. 6, 885–899. MR 2466915, DOI 10.1515/JGT.2008.057
Michael D. Fried and Moshe Jarden, Field arithmetic, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 11, Springer-Verlag, Berlin, 2008. Revised by Jarden. MR 2445111
Joel Friedman, Sheaves on graphs, their homological invariants, and a proof of the Hanna Neumann conjecture: with an appendix by Warren Dicks, Mem. Amer. Math. Soc. 233 (2015), no. 1100, xii+106. With an appendix by Warren Dicks. MR 3289057, DOI 10.1090/memo/1100
S. M. Gersten, Intersections of finitely generated subgroups of free groups and resolutions of graphs, Invent. Math. 71 (1983), no. 3, 567–591. MR 695907, DOI 10.1007/BF02095994
Leon Greenberg, Discrete groups of motions, Canadian J. Math. 12 (1960), 415–426. MR 115130, DOI 10.4153/CJM-1960-036-8
Frédéric Haglund, Finite index subgroups of graph products, Geom. Dedicata 135 (2008), 167–209. MR 2413337, DOI 10.1007/s10711-008-9270-0
Frédéric Haglund and Daniel T. Wise, Special cube complexes, Geom. Funct. Anal. 17 (2008), no. 5, 1551–1620. MR 2377497, DOI 10.1007/s00039-007-0629-4
Marshall Hall Jr., Coset representations in free groups, Trans. Amer. Math. Soc. 67 (1949), 421–432. MR 32642, DOI 10.1090/S0002-9947-1949-0032642-4
John Hempel, The finitely generated intersection property for Kleinian groups, Knot theory and manifolds (Vancouver, B.C., 1983) Lecture Notes in Math., vol. 1144, Springer, Berlin, 1985, pp. 18–24. MR 823280, DOI 10.1007/BFb0075010
A. G. Howson, On the intersection of finitely generated free groups, J. London Math. Soc. 29 (1954), 428–434. MR 65557, DOI 10.1112/jlms/s1-29.4.428
S. V. Ivanov, Intersecting free subgroups in free products of groups, Internat. J. Algebra Comput. 11 (2001), no. 3, 281–290. MR 1847180, DOI 10.1142/S0218196701000267
S. V. Ivanov, On the Kurosh rank of the intersection of subgroups in free products of groups, Adv. Math. 218 (2008), no. 2, 465–484. MR 2407942, DOI 10.1016/j.aim.2008.01.003
Andrei Jaikin-Zapirain, Approximation by subgroups of finite index and the Hanna Neumann conjecture, Duke Math. J. 166 (2017), no. 10, 1955–1987. MR 3679885, DOI 10.1215/00127094-0000015X
Moshe Jarden, A Karrass-Solitar theorem for profinite groups, J. Group Theory 9 (2006), no. 2, 139–146. MR 2220570, DOI 10.1515/JGT.2006.009
Ilya Kapovich, Amalgamated products and the Howson property, Canad. Math. Bull. 40 (1997), no. 3, 330–340. MR 1464841, DOI 10.4153/CMB-1997-039-3
A. Karrass and D. Solitar, The subgroups of a free product of two groups with an amalgamated subgroup, Trans. Amer. Math. Soc. 150 (1970), 227–255. MR 260879, DOI 10.1090/S0002-9947-1970-0260879-9
Ant. A. Klyachko and A. M. Mazhuga, Verbally closed virtually free subgroups, Mat. Sb. 209 (2018), no. 6, 75–82 (Russian, with Russian summary); English transl., Sb. Math. 209 (2018), no. 6, 850–856. MR 3807907, DOI 10.4213/sm8942
Dessislava H. Kochloukova, Subdirect products of free pro-$p$ and Demushkin groups, Internat. J. Algebra Comput. 23 (2013), no. 5, 1079–1098. MR 3096312, DOI 10.1142/S0218196713500173
Dessislava H. Kochloukova and Pavel Zalesskii, Free-by-Demushkin pro-$p$ groups, Math. Z. 249 (2005), no. 4, 731–739. MR 2126211, DOI 10.1007/s00209-004-0720-6
D. H. Kochloukova and P. A. Zalesskii, Fully residually free pro-$p$ groups, J. Algebra 324 (2010), no. 4, 782–792. MR 2651568, DOI 10.1016/j.jalgebra.2010.04.019
D. H. Kochloukova and P. A. Zalesskii, On pro-$p$ analogues of limit groups via extensions of centralizers, Math. Z. 267 (2011), no. 1-2, 109–128. MR 2772243, DOI 10.1007/s00209-009-0611-y
Jochen Koenigsmann, Pro-$p$ Galois groups of rank $\leq 4$, Manuscripta Math. 95 (1998), no. 2, 251–271. MR 1603333, DOI 10.1007/s002290050027
Serge Lang, Algebra, 3rd ed., Graduate Texts in Mathematics, vol. 211, Springer-Verlag, New York, 2002. MR 1878556, DOI 10.1007/978-1-4613-0041-0
D. D. Long and A. W. Reid, Subgroup separability and virtual retractions of groups, Topology 47 (2008), no. 3, 137–159. MR 2414358, DOI 10.1016/j.top.2006.04.001
John P. Labute, Demuškin groups of rank $\aleph _{0}$, Bull. Soc. Math. France 94 (1966), 211–244. MR 222177
John P. Labute, Classification of Demushkin groups, Canadian J. Math. 19 (1967), 106–132. MR 210788, DOI 10.4153/CJM-1967-007-8
John Labute, Nicole Lemire, Ján Mináč, and John Swallow, Demuškin groups, Galois modules, and the elementary type conjecture, J. Algebra 304 (2006), no. 2, 1130–1146. MR 2265509, DOI 10.1016/j.jalgebra.2005.12.021
Alexander Lubotzky, Combinatorial group theory for pro-$p$-groups, J. Pure Appl. Algebra 25 (1982), no. 3, 311–325. MR 666024, DOI 10.1016/0022-4049(82)90086-X
A. Martino and E. Ventura, Examples of retracts in free groups that are not the fixed subgroup of any automorphism, J. Algebra 269 (2003), no. 2, 735–747. MR 2015863, DOI 10.1016/S0021-8693(03)00535-0
O. V. Mel′nikov, Subgroups and the homology of free products of profinite groups, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), no. 1, 97–120 (Russian); English transl., Math. USSR-Izv. 34 (1990), no. 1, 97–119. MR 992980, DOI 10.1070/IM1990v034n01ABEH000607
Igor Mineyev, Submultiplicativity and the Hanna Neumann conjecture, Ann. of Math. (2) 175 (2012), no. 1, 393–414. MR 2874647, DOI 10.4007/annals.2012.175.1.11
Igor Mineyev, The topology and analysis of the Hanna Neumann conjecture, J. Topol. Anal. 3 (2011), no. 3, 307–376. MR 2831266, DOI 10.1142/S1793525311000611
Alexei G. Myasnikov and Vitaly Roman’kov, Verbally closed subgroups of free groups, J. Group Theory 17 (2014), no. 1, 29–40. MR 3176650, DOI 10.1515/jgt-2013-0034
Ján Mináč and Roger Ware, Demuškin groups of rank $\aleph _0$ as absolute Galois groups, Manuscripta Math. 73 (1991), no. 4, 411–421. MR 1134572, DOI 10.1007/BF02567651
Ján Mináč and Roger Ware, Pro-$2$-Demuškin groups of rank $\aleph _0$ as Galois groups of maximal $2$-extensions of fields, Math. Ann. 292 (1992), no. 2, 337–353. MR 1149039, DOI 10.1007/BF01444625
Hanna Neumann, On the intersection of finitely generated free groups, Publ. Math. Debrecen 4 (1956), 186–189. MR 78992
Peter Nickolas, Intersections of finitely generated free groups, Bull. Austral. Math. Soc. 31 (1985), no. 3, 339–348. MR 801592, DOI 10.1017/S0004972700009291
Jürgen Neukirch, Alexander Schmidt, and Kay Wingberg, Cohomology of number fields, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 323, Springer-Verlag, Berlin, 2008. MR 2392026, DOI 10.1007/978-3-540-37889-1
B. C. Oltikar and Luis Ribes, On prosupersolvable groups, Pacific J. Math. 77 (1978), no. 1, 183–188. MR 507629
Panagiotis A. Paramantzoglou, On the Howson property of HNN-extensions with abelian base group and amalgamated free products of abelian groups, Arch. Math. (Basel) 98 (2012), no. 2, 115–128. MR 2890286, DOI 10.1007/s00013-011-0344-0
Priyam Patel, On a theorem of Peter Scott, Proc. Amer. Math. Soc. 142 (2014), no. 8, 2891–2906. MR 3209342, DOI 10.1090/S0002-9939-2014-12031-4
Reinhold Baer, Absolute retracts in group theory, Bull. Amer. Math. Soc. 52 (1946), 501–506. MR 16419, DOI 10.1090/S0002-9904-1946-08601-2
Luis Ribes, Virtually free factors of pro-$p$ groups, Israel J. Math. 74 (1991), no. 2-3, 337–346. MR 1135243, DOI 10.1007/BF02775795
Luis Ribes, Profinite graphs and groups, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 66, Springer, Cham, 2017. MR 3677322, DOI 10.1007/978-3-319-61199-0
Luis Ribes and Pavel Zalesskii, Profinite groups, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 40, Springer-Verlag, Berlin, 2010. MR 2599132, DOI 10.1007/978-3-642-01642-4
Luis Ribes and Pavel Zalesskii, Pro-$p$ trees and applications, New horizons in pro-$p$ groups, Progr. Math., vol. 184, Birkhäuser Boston, Boston, MA, 2000, pp. 75–119. MR 1765118
Peter Scott, Subgroups of surface groups are almost geometric, J. London Math. Soc. (2) 17 (1978), no. 3, 555–565. MR 494062, DOI 10.1112/jlms/s2-17.3.555
Jean-Pierre Serre, Sur la dimension cohomologique des groupes profinis, Topology 3 (1965), 413–420 (French). MR 180619, DOI 10.1016/0040-9383(65)90006-6
Jean-Pierre Serre, Galois cohomology, Corrected reprint of the 1997 English edition, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2002. Translated from the French by Patrick Ion and revised by the author. MR 1867431, DOI 10.1007/BFb0108758
Mark Shusterman, Ascending chains of finitely generated subgroups, J. Algebra 471 (2017), 240–250. MR 3569185, DOI 10.1016/j.jalgebra.2016.09.023
Mark Shusterman, Groups with positive rank gradient and their actions, Math. Slovaca 68 (2018), no. 2, 353–360. MR 3783389, DOI 10.1515/ms-2017-0106
Mark Shusterman, Schreier’s formula for prosupersolvable groups, Internat. J. Algebra Comput. 27 (2017), no. 1, 41–48. MR 3609404, DOI 10.1142/S0218196717500035
John Howard Smith, The structure of free prosupersolvable groups, J. Algebra 83 (1983), no. 1, 256–270. MR 710597, DOI 10.1016/0021-8693(83)90147-3
Ilir Snopce and Pavel A. Zalesskii, Subgroup properties of Demushkin groups, Math. Proc. Cambridge Philos. Soc. 160 (2016), no. 1, 1–9. MR 3432326, DOI 10.1017/S0305004115000481
Teruhiko Soma, Intersection of finitely generated surface groups, J. Pure Appl. Algebra 66 (1990), no. 1, 81–95. MR 1074696, DOI 10.1016/0022-4049(90)90125-2
Teruhiko Soma, Intersection of finitely generated surface groups. II, Bull. Kyushu Inst. Tech. Math. Natur. Sci. 38 (1991), 13–22. MR 1115494
Teruhiko Soma, $3$-manifold groups with the finitely generated intersection property, Trans. Amer. Math. Soc. 331 (1992), no. 2, 761–769. MR 1042289, DOI 10.1090/S0002-9947-1992-1042289-4
Jack Sonn, Epimorphisms of Demushkin groups, Israel J. Math. 17 (1974), 176–190. MR 349636, DOI 10.1007/BF02882237
Benjamin Steinberg, On a conjecture of Karrass and Solitar, J. Group Theory 17 (2014), no. 3, 433–444. MR 3200368, DOI 10.1515/jgt-2013-0057
Gábor Tardos, On the intersection of subgroups of a free group, Invent. Math. 108 (1992), no. 1, 29–36. MR 1156384, DOI 10.1007/BF02100597
Gábor Tardos, Towards the Hanna Neumann conjecture using Dicks’ method, Invent. Math. 123 (1996), no. 1, 95–104. MR 1376247, DOI 10.1007/BF01232368
Marvin Tretkoff, Covering space proofs in combinatorial group theory, Comm. Algebra 3 (1975), 429–457. MR 417298, DOI 10.1080/00927877508822054
C. L. Tretkoff and M. D. Tretkoff, A topological proof of a theorem of Brunner and Burns about M. Hall groups, Combinatorial methods in topology and algebraic geometry (Rochester, N.Y., 1982) Contemp. Math., vol. 44, Amer. Math. Soc., Providence, RI, 1985, pp. 85–90. MR 813104, DOI 10.1090/conm/044/813104
Th. Weigel and P. A. Zalesskii, Virtually free pro-$p$ products, Israel J. Math. 221 (2017), no. 1, 425–434. MR 3705858, DOI 10.1007/s11856-017-1548-1
John S. Wilson, Profinite groups, London Mathematical Society Monographs. New Series, vol. 19, The Clarendon Press, Oxford University Press, New York, 1998. MR 1691054
Henry Wilton, Hall’s theorem for limit groups, Geom. Funct. Anal. 18 (2008), no. 1, 271–303. MR 2399104, DOI 10.1007/s00039-008-0657-8
Kay Wingberg, On Demuskin groups with involution, Ann. Sci. École Norm. Sup. (4) 22 (1989), no. 4, 555–567. MR 1026750
Alexander Zakharov, On the rank of the intersection of free subgroups in virtually free groups, J. Algebra 418 (2014), 29–43. MR 3250439, DOI 10.1016/j.jalgebra.2014.07.010
P. A. Zalesskiĭ and O. V. Mel′nikov, Subgroups of profinite groups acting on trees, Mat. Sb. (N.S.) 135(177) (1988), no. 4, 419–439, 559 (Russian); English transl., Math. USSR-Sb. 63 (1989), no. 2, 405–424. MR 942131, DOI 10.1070/SM1989v063n02ABEH003282