[MP09] Eduardo Martínez-Pedroza. Combination of quasiconvex subgroups of relatively hyperbolic groups. *Groups Geom. Dyn.*, 3(2):317–342, 2009. 37, 38

[MP12] Eduardo Martínez-Pedroza. On quasiconvexity and relatively hyperbolic structures on groups. *Geom. Dedicata*, 157:269–290, 2012. 38

[MT94] Michael L. Mihalik and Williams Towle. Quasiconvex subgroups of negatively curved groups. *J. Pure Appl. Algebra*, 95(3):297–301, 1994. 44

[MW08] Jonathan P. McCammond and Daniel T. Wise. Locally quasiconvex small-cancellation groups. *Trans. Amer. Math. Soc.*, 360(1):237–271, 2008. 37

[New68] Bill B. Newman. Some results on one-relator groups. *Bull. Amer. Math. Soc.*, 74:568–571, 1968. 37

[NR03] G. A. Niblo and L. D. Reeves. Coxeter groups act on CAT(0) cube complexes. *J. Group Theory*, 6(3):399–413, 2003. 48

[OW11] Yann Ollivier and Daniel T. Wise. Cubulating random groups at density less than 1/6. *Trans. Amer. Math. Soc.*, 363(9):4701–4733, 2011. 49

[Pic06] Mikaël Pichot. Semi-continuity of the first $\ell^2$-Betti number on the space of finitely generated groups. *Comment. Math. Helv.*, 81(3):643–652, 2006. 19

[PT11] Jesse Peterson and Andreas Thom. Group cocycles and the ring of affiliated operators. *Invent. Math.*, 185(3):561–592, 2011. 32

[PV24] Luis Paris and Olga Varghese. Narrow normal subgroups of coxeter groups and of automorphism groups of coxeter groups. *J. Group Theory*, 27(2):255–274, 2024. 37

[Rip82] Eliyahu Rips. Subgroups of small cancellation groups. *Bull. London Math. Soc.*, 14(1):45–47, 1982. 4, 44

[Sag95] Michah Sageev. Ends of group pairs and non-positively curved cube complexes. *Proc. London Math. Soc. (3)*, 71(3):585–617, 1995. 47

[Sco78] Peter Scott. Subgroups of surface groups are almost geometric. *J. London Math. Soc. (2)*, 17(3):555–565, 1978. 4, 18

[Ser80] Jean-Pierre Serre. *Trees*. Springer-Verlag, Berlin-New York, 1980. Translated from the French by John Stillwell. 20, 21, 34

[Sta68] John R. Stallings. On torsion-free groups with infinitely many ends. *Ann. of Math. (2)*, 88:312–334, 1968. 5, 32

[Sta71] John R. Stallings. *Group theory and three-dimensional manifolds*, volume 4 of *Yale Mathematical Monographs*. Yale University Press, New Haven, Conn.-London, 1971. A James K. Whittemore Lecture in Mathematics given at Yale University, 1969. 5, 32

[SW20] Rachel Skipper and Phillip Wesolek. On the Cantor-Bendixson rank of the Grigorchuk group and the Gupta-Sidki 3 group. *J. Algebra*, 555:386–405, 2020. 3

[Š76] V. L. Širvanjan. Imbedding of the group $B(\infty, n)$ in the group $B(2, n)$. *Izv. Akad. Nauk SSSR Ser. Mat.*, 40(1):190–208, 223, 1976. 13

[Wil08] Henry Wilton. Hall's theorem for limit groups. *Geom. Funct. Anal.*, 18(1):271–303, 2008. 4, 19

[Wis04] D. T. Wise. Cubulating small cancellation groups. *Geom. Funct. Anal.*, 14(1):150–214, 2004. 49
[Wis21] Daniel T. Wise. *The structure of groups with a quasiconvex hierarchy*, volume 209 of *Annals of Mathematics Studies*. Princeton University Press, Princeton, NJ, [2021] ©2021. 46, 48, 49

[Zal23] Abdul Zalloum. Effective flipping, skewering and rank rigidity for cubulated groups with factor systems, to appear in *Trans. Am. Math. Soc.*, 2023. arXiv:2305.16298 [math.GT]. 49

[Zel11] Yevhen G. Zelenyuk. *Ultrafilters and topologies on groups*, volume 50 of *De Gruyter Expositions in Mathematics*. Walter de Gruyter GmbH & Co. KG, Berlin, 2011. 13

P.A., UNIVERSITY OF BRISTOL, SCHOOL OF MATHEMATICS, BRISTOL, UK

penelope.azuelos@bristol.ac.uk

D.G., CNRS, ENS-LYON, UNITÉ DE MATHÉMATIQUES PURES ET APPLIQUÉES, 69007 LYON, FRANCE

damien.gaboriau@ens-lyon.fr