Inkang Kim

Simplicial volume of ℚ–rank one locally symmetric spaces covered by the product of ℝ–rank one symmetric spaces

We study the simplicial volume of Q-rank one locally sym- metric spaces. Let M be a Q-rank one locally symmetric space with no local direct factors locally isometric to H 2 or SL3(R)/SO3(R). We show that the simplicial volume of M is strictly positive. We also prove that this remains true for all Q-rank one locally symmetric spaces with irreducible universal cover.First, the positivity of the simplicial volume was verified for closed negatively curved manifolds by Thurston [22], Gromov [12] and Inoue and Yano [13]. It was verified for closed locally symmetric spaces covered by SL3.R/=SO3.R/ by Savage [21] and Bucher-Karlsson [3]. Indeed, Savage gave a proof of the positivity of the simplicial volume of closed locally symmetric spaces covered by SLn.R/=SOn.R/ but it turned out to be incomplete. Bucher-Karlsson made a complete proof of the same result in the case of SL3.R/=SO3.R/. Then, Lafont and Schmidt [14] showed that the simplicial volume of all closed locally symmetric spaces of non-compact type is positive, which supports the conjecture raised by Gromov.

On deformation spaces of nonuniform hyperbolic lattices

Let

Proportionality principle for the simplicial volume of families of \mathbb Q -rank 1 locally symmetric spaces

\Gamma

Volume invariant and maximal representations of discrete subgroups of Lie groups

be a nonuniform lattice acting on real hyperbolic n-space. We show that in dimension greater than or equal to 4, the volume of a representation is constant on each connected component of the representation variety of

Simplicial volume, Barycenter method, and Bounded cohomology

\Gamma

Homological and Bloch invariants for

in SO(n,1). Furthermore, in dimensions 2 and 3, there is a semialgebraic subset of the representation variety such that the volume of a representation is constant on connected components of the semialgebraic subset. Our approach gives a new proof of the local rigidity theorem for nonuniform hyperbolic lattices and the analogue of Somas theorem, which shows that the number of orientable hyperbolic manifolds dominated by a closed, connected, orientable 3-manifold is finite, for noncompact 3-manifolds.