We prove the Farrell-Jones Conjecture for mapping tori of automorphisms of virtually torsion-free hyperbolic groups. The proof uses recently developed geometric methods for establishing the Farrell-Jones Conjecture by Bartels-L{u}ck-Reich, as well as the structure theory of mapping tori by Dahmani-Krishna.
We prove the Farrell-Jones conjecture for free-by-cyclic groups. The proof uses recently developed geometric methods for establishing the Farrell-Jones Conjecture.
We prove that the Koebe circle domain conjecture is equivalent to the Weyl type problem that every complete hyperbolic surface of genus zero is isometric to the boundary of the hyperbolic convex hull of the complement of a circle domain. It provides a new way to approach the Koebes conjecture using convex geometry. Combining our result with the work of He-Schramm on the Koebe conjecture, one establishes that every simply connected non-compact polyhedral surface is discrete conformal to the complex plane or the open unit disk. The main tool we use is Schramms transboundary extremal lengths.
We give new information about the geometry of closed, orientable hyperbolic 3-manifolds with 4-free fundamental group. As an application we show that such a manifold has volume greater than 3.44. This is in turn used to show that if M is a closed orientable hyperbolic 3-manifold such that vol M < 3.44, then H_1(M;Z/2Z) has dimension at most 7.
Let $text{Mod}(S_g)$ be the mapping class group of the closed orientable surface of genus $g geq 1$. For $k geq 2$, we consider the standard $k$-sheeted regular cover $p_k: S_{k(g-1)+1} to S_g$, and analyze the liftable mapping class group $text{LMod}_{p_k}(S_g)$ associated with the cover $p_k$. In particular, we show that $text{LMod}_{p_k}(S_g)$ is the stabilizer subgroup of $text{Mod}(S_g)$ with respect to a collection of vectors in $H_1(S_g,mathbb{Z}_k)$, and also derive a symplectic criterion for the liftability of a given mapping class under $p_k$. As an application of this criterion, we obtain a normal series of $text{LMod}_{p_k}(S_g)$, which generalizes of a well known normal series of congruence subgroups in $text{SL}(2,mathbb{Z})$. Among other applications, we describe a procedure for obtaining a finite generating set for $text{LMod}_{p_k}(S_g)$ and examine the liftability of certain finite-order and pseudo-Anosov mapping classes.