Abstract
Let \(p\) and \(\ell \) be two distinct prime numbers and let \(\Gamma \) be a group. We study the asymptotic behaviour of the mod-\(\ell \) Betti numbers in \(p\)-adic analytic towers of finite index subgroups. If \(\Theta \) is a finite \(\ell \)-group of automorphisms of \(\Gamma \), our main theorem allows to lift lower bounds for the mod-\(\ell \) cohomology growth in the fixed point group \(\Gamma ^\Theta \) to lower bounds for the growth in \(\Gamma \). We give applications to \(S\)-arithmetic groups and we also obtain a similar result for cohomology with rational coefficients.
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Notes
Here “\(\gg \)“ means the inequality “\(\ge \)“ holds for large \(n\) up to a positive constant.
References
Adem, A.: Automorphisms and cohomology of discrete groups. J. Algebra 182, 721–737 (1996)
An, J., Wang, Z.: Nonabelian cohomology with coefficients in Lie groups. Trans. Am. Math. Soc. 360(6), 3019–3040 (2008)
Bergeron, N., Linnell, P., Lück, W., Sauer, R.: On the growth of Betti numbers in \(p\)-adic analytic towers. Groups Geom. Dyn. (to appear)
Borel, A.: Introduction aux groupes arithmetiques, Act. sci. ind. (1341), Hermann, Paris (1969)
Borel, A., Serre, J.-P.: Corners and arithmetic groups. Comment. Math. Helv. 48, 436–491 (1973)
Borel, A., Tits, J.: Compléments à l’article: “Groupes réductifs”. Publ. Math. Inst. Hautes Études Sci. 41, 253–276 (1972)
Borel, A., Wallach, N.: Continuous cohomology, discrete subgroups, and representations of reductive groups (2nd ed.) Math. Surveys and Mono. (67). American Mathematical Society (2000)
Bredon, G.E.: Introduction to Compact Transformations Groups. Pure and Appl. Math. (46). Academic Press Inc, New York (1972)
Bredon, G.E.: Sheaf Theory. 2nd Ed. Graduate Texts Math. 170, Springer, New York (1997)
Brown, K.S.: Cohomology of Groups, Grad. Texts Math. (87). Springer, New York (1982)
Calegari, F., Emerton, M.: Bounds for multiplicities of unitary representations of cohomological type in spaces of cusp forms. Ann. Math. 170, 1437–1446 (2009). (2)
Calegari, F., Emerton, M.: Mod-\(p\) cohomology growth in \(p\)-adic analytic towers of 3-manifolds. Groups Geom. Dyn. 5, 355–366 (2011)
Dixon, J.D., du Sautoy, M.P.F., Mann, A., Segal, D.: Analytic pro-p Groups. LMS Lect. Notes Series (157). Cambridge University Press, Cambridge (1991)
Floyd, E.E.: On periodic maps and the Euler characteristics of associated spaces. Trans. Am. Math. Soc. 72, 138–147 (1952)
Harder, G.: A Gauss-Bonnet formula for discrete arithmetically defined groups. Ann. Sci. Éc. Norm. Supér. 4(3), 409–455 (1971). (4)
Humphreys, J.E.: Introduction to Lie Algebras and Representation Theory, Grad. Texts Math. (9). Springer, New York (1972)
Jacobson, N.: A note on automorphisms of Lie algebras. Pacific J. Math. 12, 303–315 (1962)
Kionke, S., Schwermer, J.: On the growth of the first Betti number of arithmetic hyperbolic 3-manifolds. Groups Geom. Dyn. (to appear)
Lackenby, M.: Large groups, property \((\tau )\) and the homology growth of subgroups. Math. Proc. Camb. Philos. Soc. 146(3), 625–648 (2009)
Lackenby, M.: New lower bounds on subgroup growth and homology growth. Proc. Lond. Math. Soc. 98, 271–297 (2009)
Lück, W.: Approximating \(L^2\)-invariants by their finite-dimensional analogues. Geom. Funct. Anal. 4, 455–481 (1994)
Lück, W.: Approximating \(L^2\)-invariants and homology growth. Geom. Funct. Anal. 23, 622–663 (2013)
Rajan, C.S., Venkataramana, T.N.: On the first cohomology of arithmetic groups. Manuscripta math. 105, 537–552 (2001)
Rohlfs, J.: Lefschetz numbers for arithmetic groups. Cohomology of Arithmetic Groups and Automorphic Forms. Proc. Lect. Notes Math. (1447), pp. 303–313. Springer (1990)
Rohlfs, J., Speh, B.: Automorphic representations and Lefschetz numbers. Ann. Sci. Éc. Norm. Supér. 22(3), 473–499 (1989). (4)
Sarnak, P., Xue, X.: Bounds for multiplicities of automorphic representations. Duke Math. J. 64, 207–227 (1991)
Scharlau, W.: Quadratic and Hermitian Forms, Grundlehren der math. Wiss. (270), Springer, Berlin (1985)
Serre, J.-P.: Cohomologie des groupes discrets, Prospects Math. Ann. Math. Stud. 70, 77–169 (1971)
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The author would like to thank the Max Planck Institute for Mathematics in Bonn for their hospitality and support.
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Kionke, S. On lower bounds for cohomology growth in \(p\)-adic analytic towers. Math. Z. 277, 709–723 (2014). https://doi.org/10.1007/s00209-013-1273-3
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DOI: https://doi.org/10.1007/s00209-013-1273-3