We construct a compact formal 7-manifold with closed $G_2$-structure and first Betti number $b_1=1$, which does not admit any torsion-free $G_2$-structure, that is, it such the holonomy group of associated metric is subgroup $G_2$. also associative calibrated (hence volume-minimizing) 3-tori respect to this and, for each those 3-tori, we show 3-dimensional family non-trivial deformations. fibra...

Let G be a finitely generated group and G = G0 ⊇ G1 ⊇ G2 ⊇ · · · a descending chain of finite index normal subgroups of G. Given a field K, we consider the sequence b1(Gi;K) [G:Gi] of normalized first Betti numbers of Gi with coefficients in K, which we call a K-approximation for b (2) 1 (G), the first L2-Betti number of G. In this paper we address the questions of when Qapproximation and Fp-ap...

We give examples of closed hyperbolic 3-manifolds with first Betti number 2 and 3 for which no sequence of finite abelian covering spaces increases the first Betti number. For 3-manifolds M with first Betti number 2 we give a characterization in terms of some generalized self-linking numbers of M , for there to exist a family of Zn covering spaces, Mn, in which β1(Mn) increases linearly with n....

We give examples of closed hyperbolic 3-manifolds with first Betti number 2 and 3 for which no sequence of finite abelian covering spaces increases the first Betti number. For 3-manifolds M with first Betti number 2 we give a characterization in terms of some generalized self-linking numbers of M , for there to exist a family of Zn covering spaces, Mn, in which β1(Mn) increases linearly with n....