All finitely generated free-by-cyclic groups are conjugacy separable, resolving Question 19.41 of the Kourovka Notebook and implying residual finiteness of their outer automorphism groups.
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The double of a virtually compact special Gromov-hyperbolic group along a quasiconvex subgroup is virtually compact special, with a generalization to certain graphs of groups.
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Free-by-cyclic groups are conjugacy separable
All finitely generated free-by-cyclic groups are conjugacy separable, resolving Question 19.41 of the Kourovka Notebook and implying residual finiteness of their outer automorphism groups.
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Virtual specialness of the double
The double of a virtually compact special Gromov-hyperbolic group along a quasiconvex subgroup is virtually compact special, with a generalization to certain graphs of groups.