We exhibit a finite-volume cusped hyperbolic four-manifold W with a perfect circle-valued Morse function, that is a circle-valued Morse smooth function f : W → S 1 with χ(W ) critical points, all of index 2. The map f is built by extending and smoothening a combinatorial Morse function defined by Jankiewicz -Norin -Wise [14] via Bestvina -Brady theory [4].By elaborating on this construction we also prove the following facts:• There are infinitely many finite-volume hyperbolic 4-manifolds M having a handle decomposition with bounded numbers of 1-and 3-handles, so with bounded Betti numbers b 1 (M ), b 3 (M ) and rank rk(π 1 (M )). • The kernel of f * : π 1 (W ) → π 1 (S 1 ) = Z determines a geometrically infinite hyperbolic four-manifold W obtained by adding infinitely many 2-handles to a product N × [0, 1], for some cusped hyperbolic 3-manifold N . The manifold W is infinitesimally (and hence locally) rigid.• There are type-preserving representations of the fundamental groups of m036 and of the surface Σ with genus 2 and one puncture in Isom + (H 4 ) whose image is a discrete subgroup with limit set S 3 . These representations are not faithful. We do not know if they are rigid.
We prove that exactly 6 out of the 29 rational homology 3-spheres tessellated by four or less right-angled hyperbolic dodecahedra are L-spaces. The algorithm used is based on the L-space census provided by Dunfield in [Dun20], and relies on a result by Rasmussen-Rasmussen [RR17]. We use the existence of these manifolds together with a result of Martelli [Mar16a] to construct explicit examples of hyperbolic 4-manifolds containing separating L-spaces, and therefore having vanishing Seiberg-Witten invariants. This answers a question asked by Agol and Lin in [AL20].
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