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We prove that any arithmetic hyperbolic $n$-manifold of simplest type can either be geodesically embedded into an arithmetic hyperbolic $(n+1)$-manifold or its universal $mathrm{mod}~2$ Abelian cover can.
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By work of Uhlenbeck, the largest principal curvature of any least area fiber of a hyperbolic $3$-manifold fibering over the circle is bounded below by one. We give a short argument to show that, along certain families of fibered hyperbolic $3$-manifolds, there is a uniform lower bound for the maximum principal curvatures of a least area minimal surface which is greater than one.
We prove that the deformation space AH(M) of marked hyperbolic 3-manifolds homotopy equivalent to a fixed compact 3-manifold M with incompressible boundary is locally connected at minimally parabolic points. Moreover, spaces of Kleinian surface groups are locally connected at quasiconformally rigid points. Similar results are obtained for deformation spaces of acylindrical 3-manifolds and Bers slices.
Let n>2 and let M be an orientable complete finite volume hyperbolic n-manifold with (possibly empty) geodesic boundary having Riemannian volume vol(M) and simplicial volume ||M||. A celebrated result by Gromov and Thurston states that if M has empty boundary then the ratio between vol(M) and ||M|| is equal to v_n, where v_n is the volume of the regular ideal geodesic n-simplex in hyperbolic n-space. On the contrary, Jungreis and Kuessner proved that if the boundary of M is non-empty, then such a ratio is strictly less than v_n. We prove here that for every a>0 there exists k>0 (only depending on a and n) such that if the ratio between the volume of the boundary of M and the volume of M is less than k, then the ratio between vol(M) and ||M|| is greater than v_n-a. As a consequence we show that for every a>0 there exists a compact orientable hyperbolic n-manifold M with non-empty geodesic boundary such that the ratio between vol(M) and ||M|| is greater than v_n-a. Our argument also works in the case of empty boundary, thus providing a somewhat new proof of the proportionality principle for non-compact finite-volume hyperbolic n-manifolds without boundary.
Let $text{G}(n)$ be equal either to $text{PO}(n,1),text{PU}(n,1)$ or $text{PSp}(n,1)$ and let $Gamma leq text{G}(n)$ be a uniform lattice. Denote by $mathbb{H}^n_K$ the hyperbolic space associated to $text{G}(n)$, where $K$ is a division algebra over the reals of dimension $d=dim_{mathbb{R}} K$. Assume $d(n-1) geq 2$. In this paper we generalize natural maps to measurable cocycles. Given a standard Borel probability $Gamma$-space $(X,mu_X)$, we assume that a measurable cocycle $sigma:Gamma times X rightarrow text{G}(m)$ admits an essentially unique boundary map $phi:partial_infty mathbb{H}^n_K times X rightarrow partial_infty mathbb{H}^m_K$ whose slices $phi_x:mathbb{H}^n_K rightarrow mathbb{H}^m_K$ are atomless for almost every $x in X$. Then, there exists a $sigma$-equivariant measurable map $F: mathbb{H}^n_K times X rightarrow mathbb{H}^m_K$ whose slices $F_x:mathbb{H}^n_K rightarrow mathbb{H}^m_K$ are differentiable for almost every $x in X$ and such that $text{Jac}_a F_x leq 1$ for every $a in mathbb{H}^n_K$ and almost every $x in X$. The previous properties allow us to define the natural volume $text{NV}(sigma)$ of the cocycle $sigma$. This number satisfies the inequality $text{NV}(sigma) leq text{Vol}(Gamma backslash mathbb{H}^n_K)$. Additionally, the equality holds if and only if $sigma$ is cohomologous to the cocycle induced by the standard lattice embedding $i:Gamma rightarrow text{G}(n) leq text{G}(m)$, modulo possibly a compact subgroup of $text{G}(m)$ when $m>n$. Given a continuous map $f:M rightarrow N$ between compact hyperbolic manifolds, we also obtain an adaptation of the mapping degree theorem to this context.
Froyshov invariants are numerical invariants of rational homology three-spheres derived from gradings in monopole Floer homology. In the past few years, they have been employed to solve a wide range of problems in three and four-dimensional topology. In this paper, we look at connections with hyperbolic geometry for the class of minimal $L$-spaces. In particular, we study relations between Froyshov invariants and closed geodesics using ideas from analytic number theory. We discuss two main applications of our approach. First, we derive effective upper bounds for the Froyshov invariants of minimal hyperbolic $L$-spaces purely in terms of volume and injectivity radius. Second, we describe an algorithm to compute Froyshov invariants of minimal $L$-spaces in terms of data arising from hyperbolic geometry. As a concrete example of our method, we compute the Froyshov invariants for all spin$^c$ structures on the Seifert-Weber dodecahedral space. Along the way, we also prove several results about the eta invariants of the odd signature and Dirac operators on hyperbolic three-manifolds which might be of independent interest.
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