Recall that the rank of a finitely generated group is the minimal number of elements needed to generate it. M. White [Whi02] has proven that the injectivity radius of M is bounded above by some function of rank(π1(M)). Building on a technique that he introduced, we show that if M is a hyperbolic 3-manifold fibering over the circle with fiber Σg and rank(π1(M)) 6= 2g + 1, then the diameter of M ...

In recent years, the RFRS condition has been used to analyze virtual fibering in 3-manifold topology. Agol’s work shows that any 3-manifold with zero Euler characteristic satisfying the RFRS condition on its fundamental group virtually fibers over the circle. In this note we will show that a finitely generated nilpotent group is either virtually abelian or is not virtually RFRS. As a corollary,...

We study manifolds arising as spaces of sections of complex manifolds fibering over CP 1 with the normal bundle of each section isomorphic to O(k)⊗ Cn. Any hypercomplex manifold can be constructed as a space of sections of a complex manifold Z fibering over CP . The normal bundle of each section must be the sum of O(1)’s, and this suggests that interesting geometric structures can be obtained i...

Let Σg → E → Σh be a surface bundle over a surface with monodromy representation ρ : π1Σh → Mod(Σg) contained in the Torelli group Ig. In this paper we express the cup product structure in H∗(E,Z) in terms of the Johnson homomorphism τ : Ig → ∧(H1(Σg ,Z)). This is applied to the question of obtaining an upper bound on the maximal n such that p1 : E → Σh1 , . . . , pn : E → Σhn are fibering maps...

‎using variational arguments‎, ‎we prove the existence of infinitely‎ ‎many solutions to a class of $p$-biharmonic equation in‎ ‎$mathbb{r}^n$‎. ‎the existence of‎ ‎nontrivial‎ ‎solution is established under a new‎ ‎set of hypotheses on the potential $v(x)$ and the weight functions‎ ‎$h_1(x)‎, ‎h_2(x)$‎.

Abstract We show that every 3- $$(\alpha ,\delta )$$ ( α , δ ) -Sasaki manifold of dimension $$4n + 3$$ 4 n + 3 admits a locally defined Riemannian sub...