Semistability conjecture for relatively hyperbolic groups

Let GG be a finitely generated group hyperbolic relative to a finite collection {P1,,Pn}\{P_1,\ldots,P_n\} of proper finitely generated subgroups. Semistability conjecture. If each PiP_i has semistable fundamental group at \infty, then GG has semistable fundamental group at \infty. This conjecture asks whether semistability at infinity passes from peripheral subgroups to a relatively hyperbolic group; it is the primary semistability question left open after the known result for relatively hyperbolic groups whose boundary has no cut point.

Sources & referencesView supporting material

Primary source

Matthew Haulmark and Michael Mihalik, “Relatively Hyperbolic Groups with Semistable Peripheral Subgroups”, arXiv:2101.05923 (2021).

Additional references

2 papers in this index state this conjecture (2012–2021). The statement above is taken from the most recent of them; the others are arXiv:1210.6741.

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