\(\text{CAT}(-1)\) structures for free-by-free groups (Q1597479)

It is known that a group which acts discretely and cocompactly by isometries on a \(\text{CAT}(-1)\) space (a generalization of simply connected manifolds of negative sectional curvatures) is hyperbolic (as defined by Gromov by placing restrictions on the large scale geometry of geodesic triangles in the group), but it is not known if every hyperbolic group is \(\text{CAT}(-1)\). Examples of free-by-free hyperbolic groups (extensions of free by free groups) are known by work of Bestvina, Feighn and Handel, and examples of \(\text{CAT}(-1)\) free-by-cyclic groups were constructed by McCammond and by Wise. In the present paper, the first examples of \(\text{CAT}(-1)\) free-by-free groups are constructed, using tree valued Morse theory and branched coverings of 2-complexes; the construction produces also examples of noncoherent hyperbolic groups.