The existence of a hyperbolic group which is not residually finite is (to my knowledge) an open question. Is there any reason to suspect that all hyperbolic groups are residually finite, perhaps some hope that existing techniques for showing residual finiteness can be extended to all hyperbolic groups? Alternatively, is it generally suspected that hyperbolic groups which are not residually finite exist but techniques to construct them are unknown?
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1$\begingroup$ I don't think that there is a general belief in either direction (well, I heard some say so in the direction of the existence of non-RF hyperbolic groups, but I don't share this impression). However, I think that if it's ever true that all hyperbolic groups are RF, this doesn't follow from "existing techniques". $\endgroup$– YCorCommented Mar 10, 2023 at 15:56
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5$\begingroup$ Here are generalities you may know. Sela proved that every torsion-free hyperbolic group is Hopfian, and being Hopfian is a necessary condition for being RF (by Malcev). Gromov more or less conjectured that not all hyperbolic groups are RF in his monograph on hyperbolic groups. If all hyperbolic groups are RF, then it follows from a theorem of Olshanskii that for every non-elementary hyperbolic group $G$, every finite group embeds in a finite quotient of $G$ ("S-finite-Q-universality"). So RF for hyperbolic groups means that there really are lots of finite quotients. [contd.] $\endgroup$– Carl-Fredrik Nyberg BroddaCommented Mar 10, 2023 at 17:06
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3$\begingroup$ [contd.] Assuming that every hyperbolic group is RF, here are two more corollaries: every hyperbolic group is virtually torsion-free (Kapovich-Wise); every quasi-convex subgroup of every hyperbolic group is separable (Agol-Groves-Manning). This latter would show that if hyperbolic groups are all RF, then we have even more finite quotients. Finally, some techniques have been ruled out for constructing non-residually finite hyperbolic groups, e.g. in this nice paper by Alan Logan. $\endgroup$– Carl-Fredrik Nyberg BroddaCommented Mar 10, 2023 at 17:12
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1$\begingroup$ Regarding extending existing techniques: One common method of proving residual finiteness of hyperbolic groups is cubulation. For example, one-relator groups with torsion and small cancellation groups were shown to be residually finite by proving that a finite index subgroup acts on a CAT(0) cubical complex in a specific (special...) way. However, cubulated hyperbolic groups are always linear, and M. Kapovich proved that there are non-linear hyperbolic groups. Therefore, cubulation cannot be extended to all hyperbolic groups. $\endgroup$– ADLCommented Mar 11, 2023 at 10:31
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1$\begingroup$ @GilesGardam In his monograph, p. 141, in a remark: for a group $\Gamma$ he discusses sequences of subgroups $\Gamma_1 \supset \Gamma_2 \supset \cdots$ of finite index, and when such a sequence satisfies the condition $\bigcap_i \Gamma_i = 1$. He then says "Probably, 'generic' word hyperbolic groups admit no such sequences of subgroups of finite index". $\endgroup$– Carl-Fredrik Nyberg BroddaCommented Oct 4 at 15:16
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