A new proof of finiteness of maximal arithmetic reflection groups
Annales Henri Lebesgue, Volume 6 (2023), pp. 151-159.


Keywords Reflection groups, hyperbolic geometry


We give a new proof of the finiteness of maximal arithmetic reflection groups. Our proof is novel in that it makes no use of trace formulas or other tools from the theory of automorphic forms and instead relies on the arithmetic Margulis lemma of Fraczyk, Hurtado and Raimbault.


[ABSW08] Agol, Ian; Belolipetsky, Mikhail; Storm, Peter; Whyte, Kevin Finiteness of arithmetic hyperbolic reflection groups, Groups Geom. Dyn., Volume 2 (2008) no. 4, pp. 481-498 | DOI | MR | Zbl

[Ago06] Agol, Ian Finiteness of arithmetic Kleinian reflection groups, Proceedings of the international congress of mathematicians (ICM), Madrid, Spain, August 22–30, 2006. Volume II: Invited lectures, European Mathematical Society, 2006, pp. 951-960 | MR | Zbl

[And71] Andreev, E. M. Intersection of plane boundaries of a polytope with acute angles, Math. Notes, Volume 8 (1971), pp. 761-764 | DOI | Zbl

[Bel16] Belolipetsky, Mikhail Arithmetic hyperbolic reflection groups, Bull. Am. Math. Soc., Volume 53 (2016) no. 3, pp. 437-475 | DOI | MR | Zbl

[Bre11] Breuillard, Emmanuel A height gap theorem for finite subsets of GL d (Q ¯) and nonamenable subgroups, Ann. Math., Volume 174 (2011) no. 2, pp. 1057-1110 | DOI | MR | Zbl

[CHL21] Chen, Lvzhou; Hurtado, Sebastian; Lee, Homin A height gap in GL d (Q ¯) and almost laws (2021) (https://arxiv.org/abs/2110.15404)

[Lin18] Linowitz, Benjamin Bounds for arithmetic hyperbolic reflection groups in dimension 2, Transform. Groups, Volume 23 (2018) no. 3, pp. 743-753 | DOI | MR | Zbl

[LMR06] Long, Darren D.; Maclachlan, Colin; Reid, Alan W. Arithmetic Fuchsian groups of genus zero, Pure Appl. Math. Q., Volume 2 (2006) no. 2, pp. 569-599 | DOI | MR | Zbl

[MHR22] Mikolaj, Fraczyk; Hurtado, Sebastian; Raimbault, Jean Homotopy type and homology versus volume for arithmetic locally symmetric spaces (2022) (https://arxiv.org/abs/2202.13619)

[Nik80] Nikulin, Vyacheslav V. On the arithmetic groups generated by reflections in Lobachevsky spaces, Izv. Akad. Nauk SSSR, Ser. Mat., Volume 44 (1980) no. 3, pp. 637-669 | MR | Zbl

[Nik07] Nikulin, Vyacheslav V. Finiteness of the number of arithmetic groups generated by reflections in Lobachevsky spaces, Izv. Math., Volume 71 (2007) no. 1, pp. 53-56 translation from Izv. Ross. Akad. Nauk, Ser. Mat. 71, No. 1, pp. 55-60 (2007) | DOI | Zbl

[Rai22] Raimbault, Jean Coxeter polytopes and Benjamini–Schramm convergence (2022) (https://arxiv.org/abs/2209.03002)

[Vin85] Vinberg, Ernest B. Hyperbolic reflection groups, Russ. Math. Surv., Volume 40 (1985) no. 1, pp. 31-75 | DOI | MR | Zbl