Cohomology of group theoretic Dehn fillings. II. (English) Zbl 1536.20057
For the study of of \(3\)-manifolds, W. P. Thurston [Bull. Am. Math. Soc., New Ser. 6, 357–379 (1982; Zbl 0496.57005)] introduced the notion of a Dehn surgery, which is a two-step procedure of modifying a \(3\)-manifold. The second step of this surgery, called Dehn filling.
A celebrated result of Thurston [loc. cit.] asserts that most Dehn fillings preserve hyperbolicity.
There is an analogous construction in group theory, called (group theoretic) Dehn filling, which can be formalized as follows: Given a group \(G\) with a subgroup \(H\) and a normal subgroup \(N\) of \(H\), the Dehn filling associated with the triple \((G, H, N)\) is the quotient \(G/\langle \langle N \rangle \rangle\), where \(\langle \langle N \rangle \rangle\) is the normal closure of \(N\) in \(G\).
Algebraic analogs of Thurston’s results can be proved for groups satisfying certain negative curvature conditions. The first result of this kind was for relatively hyperbolic groups by D. V. Osin [Invent. Math. 167, No. 2, 295–326 (2007; Zbl 1116.20031)] and independently, by D. Groves and J. F. Manning [Isr. J. Math. 168, 317–429 (2008; Zbl 1211.20038)]. Later, F. Dahmani et al. [Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces. Providence, RI: American Mathematical Society (AMS) (2016; Zbl 1396.20041)] introduced a generalization of relative hyperbolicity based on the notion of a hyperbolically embedded subgroup and proved a generalization of the main results in the above mentioned papers.
That the subgroup \(H\) of the group \(G\) is hyperbolically embedded is denoted by \(H \hookrightarrow _{h} G\). For the definition and motivation of hyperbolically embedded subgroups we refer to paragraph 3.3 in the paper.
The authors motivated by Thurston’s results asked the following:
Question. For a group \(G\) with a subgroup \(H\hookrightarrow _{h} G\) and a normal subgroup \(N\) of \(H\), what can be said about the cohomology of \(G/\langle \langle N \rangle \rangle\)?
The main goal of this paper (and the previous one [the second author, J. Algebra 542, 277–307 (2020; Zbl 1456.20053)]) is to address this question and to illustrate the implications of the results in this direction.
The obtained results are included in Theorems A, B, C, D, E. It is not possible to quote here the statements of these theorems.
Nevertheless, here we comment on these results. Instead of proving Theorem A, more general results are proved in Section 4, which cover the case of a hyperbolically embedded family of subgroups and are useful in the proof of Theorems B, C, and E and they also cover the case of weakly hyperbolically embedded subgroups and they can be applied to graph of groups.
Theorems B and C can be seen as natural generalizations and group theoretic analogs of the results of K. Fujiwara and J. F. Manning [J. Differ. Geom. 85, No. 2, 229–270 (2010; Zbl 1211.53066)].
As an application of Theorem D, \(SQ\)-universality of hyperbolic groups given by A. Yu. Ol’shanskij [Sb. Math. 186, No. 8, 1199–1211 (1995; Zbl 0864.20023); translation from Mat. Sb. 186, No. 8, 119–132 (1995)] is strengtened and independently by T. Delzant [Duke Math. J. 83, No. 3, 661–682 (1996; Zbl 0852.20032)] by adding cohomological conditions.
The benefit of Theorems D and E is that they allow one to control the cohomology of the resulting acylindrically hyperbolic quotients.
The notion of an acylindrically hyperbolic group was introduced by D. Osin [Trans. Am. Math. Soc. 368, No. 2, 851–888 (2016; Zbl 1380.20048)] as a generalization of non-elementary hyperbolic and non-elementary relatively hyperbolic groups. As it is illustrated (Section 8), this facilitates the constructions of various acylindrically hyperbolic groups satisfying certain cohomological properties.
A celebrated result of Thurston [loc. cit.] asserts that most Dehn fillings preserve hyperbolicity.
There is an analogous construction in group theory, called (group theoretic) Dehn filling, which can be formalized as follows: Given a group \(G\) with a subgroup \(H\) and a normal subgroup \(N\) of \(H\), the Dehn filling associated with the triple \((G, H, N)\) is the quotient \(G/\langle \langle N \rangle \rangle\), where \(\langle \langle N \rangle \rangle\) is the normal closure of \(N\) in \(G\).
Algebraic analogs of Thurston’s results can be proved for groups satisfying certain negative curvature conditions. The first result of this kind was for relatively hyperbolic groups by D. V. Osin [Invent. Math. 167, No. 2, 295–326 (2007; Zbl 1116.20031)] and independently, by D. Groves and J. F. Manning [Isr. J. Math. 168, 317–429 (2008; Zbl 1211.20038)]. Later, F. Dahmani et al. [Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces. Providence, RI: American Mathematical Society (AMS) (2016; Zbl 1396.20041)] introduced a generalization of relative hyperbolicity based on the notion of a hyperbolically embedded subgroup and proved a generalization of the main results in the above mentioned papers.
That the subgroup \(H\) of the group \(G\) is hyperbolically embedded is denoted by \(H \hookrightarrow _{h} G\). For the definition and motivation of hyperbolically embedded subgroups we refer to paragraph 3.3 in the paper.
The authors motivated by Thurston’s results asked the following:
Question. For a group \(G\) with a subgroup \(H\hookrightarrow _{h} G\) and a normal subgroup \(N\) of \(H\), what can be said about the cohomology of \(G/\langle \langle N \rangle \rangle\)?
The main goal of this paper (and the previous one [the second author, J. Algebra 542, 277–307 (2020; Zbl 1456.20053)]) is to address this question and to illustrate the implications of the results in this direction.
The obtained results are included in Theorems A, B, C, D, E. It is not possible to quote here the statements of these theorems.
Nevertheless, here we comment on these results. Instead of proving Theorem A, more general results are proved in Section 4, which cover the case of a hyperbolically embedded family of subgroups and are useful in the proof of Theorems B, C, and E and they also cover the case of weakly hyperbolically embedded subgroups and they can be applied to graph of groups.
Theorems B and C can be seen as natural generalizations and group theoretic analogs of the results of K. Fujiwara and J. F. Manning [J. Differ. Geom. 85, No. 2, 229–270 (2010; Zbl 1211.53066)].
As an application of Theorem D, \(SQ\)-universality of hyperbolic groups given by A. Yu. Ol’shanskij [Sb. Math. 186, No. 8, 1199–1211 (1995; Zbl 0864.20023); translation from Mat. Sb. 186, No. 8, 119–132 (1995)] is strengtened and independently by T. Delzant [Duke Math. J. 83, No. 3, 661–682 (1996; Zbl 0852.20032)] by adding cohomological conditions.
The benefit of Theorems D and E is that they allow one to control the cohomology of the resulting acylindrically hyperbolic quotients.
The notion of an acylindrically hyperbolic group was introduced by D. Osin [Trans. Am. Math. Soc. 368, No. 2, 851–888 (2016; Zbl 1380.20048)] as a generalization of non-elementary hyperbolic and non-elementary relatively hyperbolic groups. As it is illustrated (Section 8), this facilitates the constructions of various acylindrically hyperbolic groups satisfying certain cohomological properties.
Reviewer: Dimitrios Varsos (Athína)
MSC:
| 20F67 | Hyperbolic groups and nonpositively curved groups |
| 20F10 | Word problems, other decision problems, connections with logic and automata (group-theoretic aspects) |
| 20E06 | Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations |
Keywords:
cohomology of groups; Dehn filling; hyperbolically embedded subgroup; Cohen-Lyndon property; SQ-universality; Poincaré duality; simplicial volume; cohomological finiteness conditionsCitations:
Zbl 0496.57005; Zbl 1116.20031; Zbl 1211.20038; Zbl 1396.20041; Zbl 1456.20053; Zbl 0864.20023; Zbl 0852.20032; Zbl 1380.20048; Zbl 1211.53066References:
| [1] | Antolín, Y.; Coulon, R.; Gandini, G., Farrell-Jones via Dehn fillings. J. Topol. Anal., 4, 873-895 (2018) · Zbl 1451.57011 |
| [2] | Agol, I.; Groves, D.; Manning, J., An alternate proof of Wise’s malnormal special quotient theorem. Forum Math. Pi (2016) · Zbl 1380.20047 |
| [3] | Agol, I., The virtual Haken conjecture. Doc. Math., 1045-1087 (2013), With an appendix by I. Agol, D. Groves, and J. Manning · Zbl 1286.57019 |
| [4] | Alonso, J., Finiteness conditions on groups and quasi-isometries. J. Pure Appl. Algebra, 2, 121-129 (1994) · Zbl 0823.20034 |
| [5] | Arenas, M., A cubical rips construction (2022) |
| [6] | Bieri, R.; Eckmann, B., Relative homology and Poincaré duality for group pairs. J. Pure Appl. Algebra, 3, 277-319 (1978) · Zbl 0392.20032 |
| [7] | Bestvina, M., Questions in geometric group theory |
| [8] | Bestvina, M.; Feighn, M., A hyperbolic \(O u t( F_n)\)-complex. Groups Geom. Dyn., 1, 31-58 (2010) · Zbl 1190.20017 |
| [9] | Brown, K.; Geoghegan, R., An infinite-dimensional torsion-free \(F P_\infty\) group. Invent. Math., 2, 367-381 (1984) · Zbl 0557.55009 |
| [10] | Bleiler, S. A.; Hodgson, C. D., Spherical space forms and Dehn filling. Topology, 3, 809-833 (1996) · Zbl 0863.57009 |
| [11] | Bieri, R., Mayer-Vietoris sequences for HNN-groups and homological duality. Math. Z., 2, 123-130 (1975) · Zbl 0407.20040 |
| [12] | Bieri, R., Homological Dimension of Discrete Groups. Queen Mary College Mathematical Notes (1981), Queen Mary College, Department of Pure Mathematics: Queen Mary College, Department of Pure Mathematics London |
| [13] | Bowditch, B., Tight geodesics in the curve complex. Invent. Math., 2, 281-300 (2008) · Zbl 1185.57011 |
| [14] | Bowditch, B., Relatively hyperbolic groups. Int. J. Algebra Comput., 3 (2012) · Zbl 1259.20052 |
| [15] | Brady, N., Branched coverings of cubical complexes and subgroups of hyperbolic groups. J. Lond. Math. Soc. (2), 2, 461-480 (1999) · Zbl 0940.20048 |
| [16] | Bredon, G. E., Topology and Geometry. Graduate Texts in Mathematics (1993), Springer-Verlag: Springer-Verlag New York · Zbl 0791.55001 |
| [17] | Brown, K., Homological criteria for finiteness. Comment. Math. Helv., 129-135 (1975) · Zbl 0302.18010 |
| [18] | Brown, K., Finiteness properties of groups, 45-75 · Zbl 0613.20033 |
| [19] | Brown, K., Cohomology of Groups. Graduate Texts in Mathematics (1994), Springer-Verlag: Springer-Verlag New York, Corrected reprint of the 1982 original |
| [20] | Cartan, H.; Eilenberg, S., Homological Algebra (1956), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 0075.24305 |
| [21] | Cohen, D.; Lyndon, R., Free bases for normal subgroups of free groups. Trans. Am. Math. Soc., 526-537 (1963) · Zbl 0115.25103 |
| [22] | Cantat, S.; Lamy, S., Normal subgroups in the Cremona group. Acta Math., 1, 31-94 (2013), With an appendix by Yves de Cornulier · Zbl 1278.14017 |
| [23] | Degrijse, D., Amenable groups of finite cohomological dimension and the zero divisor conjecture. Math. Ann., 1-2, 1151-1162 (2023) · Zbl 1521.20113 |
| [24] | Dekimpe, K., Almost-Bieberbach Groups: Affine and Polynomial Structures. Lecture Notes in Mathematics (1996), Springer-Verlag: Springer-Verlag Berlin · Zbl 0865.20001 |
| [25] | Delzant, T., Sous-groupes distingués et quotients des groupes hyperboliques. Duke Math. J., 3, 661-682 (1996) · Zbl 0852.20032 |
| [26] | Dahmani, F.; Guirardel, V., Recognizing a relatively hyperbolic group by its Dehn fillings. Duke Math. J., 12, 2189-2241 (2018) · Zbl 1511.20153 |
| [27] | Dahmani, F.; Guirardel, V.; Osin, D., Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces. Mem. Am. Math. Soc., 1156 (2017), v+152 |
| [28] | Farb, B., Relatively hyperbolic groups. Geom. Funct. Anal., 5, 810-840 (1998) · Zbl 0985.20027 |
| [29] | Fujiwara, K.; Manning, J. F., \( \operatorname{CAT}(0)\) and \(\operatorname{CAT}(- 1)\) fillings of hyperbolic manifolds. J. Differ. Geom., 2, 229-269 (2010) |
| [30] | Fujiwara, K.; Manning, J. F., Simplicial volume and fillings of hyperbolic manifolds. Algebraic Geom. Topol., 4, 2237-2264 (2011) · Zbl 1244.53051 |
| [31] | Franceschini, F., A characterization of relatively hyperbolic groups via bounded cohomology. Groups Geom. Dyn., 3, 919-960 (2018) · Zbl 1456.20052 |
| [32] | Frigerio, R., Bounded Cohomology of Discrete Groups. Mathematical Surveys and Monographs (2017), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 1398.57003 |
| [33] | Groves, D.; Manning, J., Dehn filling in relatively hyperbolic groups. Isr. J. Math., 317-429 (2008) · Zbl 1211.20038 |
| [34] | Groves, D.; Manning, J., Dehn fillings and elementary splittings. Trans. Am. Math. Soc., 5, 3017-3051 (2018) · Zbl 1427.20058 |
| [35] | Groves, D.; Manning, J. F.; Sisto, A., Boundaries of Dehn fillings. Geom. Topol., 6, 2929-3002 (2019) · Zbl 1469.20042 |
| [36] | Grothendieck, A., Sur quelques points d’algèbre homologique. Tohoku Math. J. (2), 2, 119-221 (1957) · Zbl 0118.26104 |
| [37] | Gromov, M., Almost flat manifolds. J. Differ. Geom., 2, 231-241 (1978) · Zbl 0432.53020 |
| [38] | Gromov, M., Volume and bounded cohomology. Inst. Hautes Études Sci. Publ. Math., 5-99 (1983), 1982 · Zbl 0516.53046 |
| [39] | Gromov, M., Asymptotic invariants of infinite groups, 1-295 |
| [40] | Gruber, D.; Sisto, A., Infinitely presented graphical small cancellation groups are acylindrically hyperbolic. Ann. Inst. Fourier (Grenoble), 6, 2501-2552 (2018) · Zbl 1483.20059 |
| [41] | Hochschild, G.; Serre, J., Cohomology of group extensions. Trans. Am. Math. Soc., 110-134 (1953) · Zbl 0050.02104 |
| [42] | Hull, M., Small cancellation in acylindrically hyperbolic groups. Groups Geom. Dyn., 4, 1077-1119 (2016) · Zbl 1395.20028 |
| [43] | Italiano, G.; Martelli, B.; Migliorini, M., Hyperbolic 5-manifolds that fiber over \(S^1\). Invent. Math., 1, 1-38 (2023) · Zbl 1512.57039 |
| [44] | Ivanov, N. V., Foundations of the theory of bounded cohomology, 69-109, 177-178 · Zbl 0573.55007 |
| [45] | Jitsukawa, T., Malnormal subgroups of free groups, 83-95 · Zbl 1018.20018 |
| [46] | Jankiewicz, K.; Norin, S.; Wise, D., Virtually fibering right-angled Coxeter groups. J. Inst. Math. Jussieu, 3, 957-987 (2021) · Zbl 1508.20045 |
| [47] | Johnson, F. E.A.; Wall, C. T.C., On groups satisfying Poincaré duality. Ann. Math. (2), 592-598 (1972) · Zbl 0245.57005 |
| [48] | Löh, C., Simplicial volume. Bull. Manifold Atlas (2011) |
| [49] | Lamy, S.; Przytycki, P., Acylindrical hyperbolicity of the three-dimensional tame automorphism group. Ann. Sci. Éc. Norm. Supér. (4), 2, 367-392 (2019) · Zbl 1442.14188 |
| [50] | Lyndon, R.; Schupp, P., Combinatorial Group Theory. Classics in Mathematics (2001), Springer-Verlag: Springer-Verlag Berlin, Reprint of the 1977 edition |
| [51] | Lyndon, R., The cohomology theory of group extensions. Duke Math. J., 271-292 (1948) · Zbl 0031.19802 |
| [52] | Lyndon, R., Cohomology theory of groups with a single defining relation. Ann. Math. (2), 650-665 (1950) · Zbl 0039.02302 |
| [53] | Masur, H.; Minsky, Y., Geometry of the complex of curves. I. Hyperbolicity. Invent. Math., 1, 103-149 (1999) · Zbl 0941.32012 |
| [54] | Maher, J.; Sisto, A., Random subgroups of acylindrically hyperbolic groups and hyperbolic embeddings. Int. Math. Res. Not., 3941-3980 (2019) · Zbl 1457.20032 |
| [55] | Mineyev, I.; Yaman, A., Relative hyperbolicity and bounded cohomology (2007), preprint |
| [56] | Ollivier, Y., Sharp phase transition theorems for hyperbolicity of random groups. Geom. Funct. Anal., 3, 595-679 (2004) · Zbl 1064.20045 |
| [57] | Ollivier, Y., A January 2005 Invitation to Random Groups. Ensaios Matemáticos. Mathematical Surveys (2005), Sociedade Brasileira de Matemática: Sociedade Brasileira de Matemática Rio de Janeiro · Zbl 1163.20311 |
| [58] | Olshanskii, A. Yu., SQ-universality of hyperbolic groups. Mat. Sb., 8, 119-132 (1995) |
| [59] | Osin, D., Relatively hyperbolic groups: intrinsic geometry, algebraic properties, and algorithmic problems. Mem. Am. Math. Soc., 843 (2006), vi+100 · Zbl 1093.20025 |
| [60] | Osin, D., Peripheral fillings of relatively hyperbolic groups. Invent. Math., 2, 295-326 (2007) · Zbl 1116.20031 |
| [61] | Osin, D., Questions on relatively hyperbolic groups and related classes (2008) |
| [62] | Osin, D., Acylindrically hyperbolic groups. Trans. Am. Math. Soc., 2, 851-888 (2016) · Zbl 1380.20048 |
| [63] | Osin, D., Groups acting acylindrically on hyperbolic spaces, 915-936 |
| [64] | Osin, D., A topological zero-one law and elementary equivalence of finitely generated groups. Ann. Pure Appl. Log., 3 (2021) · Zbl 1498.03085 |
| [65] | Ruh, E. A., Almost flat manifolds. J. Differ. Geom., 1, 1-14 (1982) · Zbl 0468.53036 |
| [66] | Sun, B., A dynamical characterization of acylindrically hyperbolic groups. Algebraic Geom. Topol., 4, 1711-1745 (2019) · Zbl 1481.20162 |
| [67] | Sun, B., Cohomology of group theoretic Dehn fillings I: Cohen-Lyndon type theorems. J. Algebra, 277-307 (2020) · Zbl 1456.20053 |
| [68] | Swan, R., Groups of cohomological dimension one. J. Algebra, 585-610 (1969) · Zbl 0188.07001 |
| [69] | Thurston, W., Three-dimensional manifolds, Kleinian groups and hyperbolic geometry. Bull. Am. Math. Soc. (N.S.), 3, 357-381 (1982) · Zbl 0496.57005 |
| [70] | (Wall, C. T.C., Homological Group Theory. Homological Group Theory, London Mathematical Society Lecture Note Series, vol. 36 (1979), Cambridge University Press: Cambridge University Press Cambridge-New York) · Zbl 0409.00004 |
| [71] | Wang, O. H., A spectral sequence for Dehn fillings. Algebraic Geom. Topol., 5, 2257-2272 (2021) · Zbl 07432510 |
| [72] | Weibel, C. A., An Introduction to Homological Algebra. Cambridge Studies in Advanced Mathematics (1994), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0797.18001 |
| [73] | Żuk, A., Property (T) and Kazhdan constants for discrete groups. Geom. Funct. Anal., 3, 643-670 (2003) · Zbl 1036.22004 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
