Abstract
We study when the fiber product of groups with Property (T) has Property (T).
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References
Bass H and Lubotzky A, Non-arithmetic super-rigid groups: Counterexamples to Platonov’s conjecture, Ann. Math. 151 (2000) 1151–1173
Bekka B, de la Harpe P and Valette A, Kazhdan’s Property (T) (2008) (Cambridge University Press)
Biswas I, Koberda T, Mj M and Santharoubane R, Representations of surface groups with finite mapping class group orbits, New York J. Math. 24 (2018) 241–250
Bridson M R and Miller III C F, Structure and finiteness properties of subdirect products of groups, Proc. London Math. Soc. (3)98 (2009) 631–651
Burger M, Kazhdan constants for \(\text{SL}_{3}(\mathbb{Z} )\), J. Reine Angew. Math. 431 (1991) 36–67
Chatterji I, Witte Morris D and Shah R, Relative property (T) for nilpotent subgroups, Doc. Math. 23 (2018) 353–382
Chevalley C and Weil A, Über das Verhalten der Integrale 1. Gattung bei Automorphismen des Funktionenkörpers, Abh. Math. Sem. Univ. Hamburg 10 (1934) 358–361
de Cornulier Y, Relative Kazhdan property, Ann. Sci. École Norm. Sup. (4) 39 (2006) 301–333
de Cornulier Y ,Tessera R, A characterization of relative Kazhdan property T for semidirect products with abelian groups, Ergod. Th. & Dynam. Sys. 31 (2011) 793–805
Dahmani F, Guirardel V and Osin D, Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces, Mem. Amer. Math. Soc. 245 (1156) (2017) v+152
de la Harpe P and Valette A, La propriété (T) de Kazhdan pour les groupes localement compacts, Astérisque 175 (1989)
Folland G B, A course in abstract harmonic analysis, second edition (2016) (CRC Press)
Gromov M, Hyperbolic groups, Essays in group theory, 75–263, Math. Sci. Res. Inst. Publ. 8 (1987) (NewYork: Springer)
Grunewald F, Larsen M, Lubotzky A and Malestein J, Arithmetic quotients of the mapping class group, Geom. Funct. Anal. 25 (2015) 1493–1542
Ioana A, Relative property (T) for the subequivalence relations induced by the action of \(\text{ SL}_2(Z)\) on \({\mathbb{T}}^2\), Adv. Math. 224 (2010) 1589–1617
Jolissaint P, On property (T) for pairs of topological groups, Enseign. Math. (2) 51 (2005) 31–45
Kazhdan D A, On the connection of the dual space of a group with the structure of its closed subgroups (Russian), Funkcional. Anal. i Priložen. 1 (1967) 71–74
Koberda T, Asymptotic linearity of the mapping class group and a homological version of the Nielsen–Thurston classification, Geom. Dedicata 156 (2012) 13–30
Margulis G A, Finitely-additive invariant measures on Euclidean spaces, Ergodic Theory Dynam. Systems 2 (1982) 383–396
Margulis G A, Discrete subgroups of semisimple lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 17 (1991) (Berlin: Springer-Verlag) x \(+\) 388 pp.
Raja C R E, Strong relative property (T) and spectral gap of random walks, Geom. Dedicata 164 (2013) 9–25
Shalom Y, Invariant measures for algebraic actions, Zariski dense subgroups and Kazhdan’s property (T), Trans. Amer. Math. Soc. 351 (1999) 3387–3412
Shalom Y, Bounded generation and Kazhdan’s property (T), Publ. Math. Inst. Hautes Études Sci. 90 (1999) 145–168
Shalom Y, Rigidity of commensurators and irreducible lattices, Invent. Math. 141 (2000) 1–54
Sun B, Cohomology of group theoretic Dehn fillings I: Cohen–Lyndon type theorems, J. Algebra 542 (2020) 277–307
Acknowledgements
The authors would like to thank Francois Dahmani for explaining a simple geometric proof of Theorem 3.10. They would also like to thank Indira Chatterji for helpful comments on an earlier draft. In particular, she asked a version of Question 3.12 when H is assumed to have the Haagerup property. Special thanks are due to the referee for a careful reading and several very helpful comments. Both authors are supported by the Department of Atomic Energy, Government of India, under Project No. 12-R &D-TFR-14001. The first author is supported in part by the Department of Science and Technology JC Bose Fellowship, and an endowment of the Infosys Foundation via the Chandrasekharan–Infosys Virtual Centre for Random Geometry.
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Mj, M., Mondal, A. Property (T) for fiber products. Proc Math Sci 132, 42 (2022). https://doi.org/10.1007/s12044-022-00687-2
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DOI: https://doi.org/10.1007/s12044-022-00687-2