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   Genevois recently classified which graph braid groups are word hyperbolic. In the 3-strand case, he asked whether all such word hyperbolic groups are actually free; this reduced to checking two infinite classes of graphs: sun and pulsar graphs. We prove that 3-strand braid groups of sun graphs are free. On the other hand, it was known to experts that 3-strand braid groups of most pulsar graphs contain surface subgroups. We provide a simple proof of this and prove an additional structure theorem for these groups. 

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   https://doi.org/10.1515/crelle-2023-0069  

   Bamler, Richard H; Kleiner, Bruce (October 2023, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract LetXbe acompact orientable non-Haken 3-manifold modeled on the Thurston geometry $\mathrm{Nil}${\operatorname{Nil}}. We show that the diffeomorphism group $\mathrm{Diff}\left(X\right)${\operatorname{Diff}(X)}deformation retracts to the isometry group $\mathrm{Isom}\left(X\right)${\operatorname{Isom}(X)}. Combining this with earlier work by many authors, this completes the determination the homotopy type of $\mathrm{Diff}\left(X\right)${\operatorname{Diff}(X)}for any compact, orientable, prime 3-manifoldX. 

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5. Ricci flow and diffeomorphism groups of 3-manifolds

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   Bamler, Richard; Kleiner, Bruce (August 2022, Journal of the American Mathematical Society)

   We complete the proof of the Generalized Smale Conjecture, apart from the case of R P 3 RP^3 , and give a new proof of Gabai’s theorem for hyperbolic 3 3 -manifolds. We use an approach based on Ricci flow through singularities, which applies uniformly to spherical space forms, except S 3 S^3 and R P 3 RP^3 , as well as hyperbolic manifolds, to prove that the space of metrics of constant sectional curvature is contractible. As a corollary, for such a 3 3 -manifold X X , the inclusion Isom ⁡ ( X , g ) → Diff ⁡ ( X ) \operatorname {Isom}(X,g)\rightarrow \operatorname {Diff}(X) is a homotopy equivalence for any Riemannian metric g g of constant sectional curvature. 

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