An introduction to ideal simplicial volume

Ora Inizio: 
Ora Fine: 
Marco Moraschini
Università di Pisa

The simplicial volume is a homotopy invariant of compact manifolds introduced in 1982 by Gromov
in his pioneering paper "Volume and Bounded Cohomology". Roughly speaking, the simplicial volume
measures how difficult is to describe a manifold in terms of real singular chains.
In this talk, we will define the ideal simplicial volume, a variation of the ordinary simplicial volume for
compact manifolds with boundary. The main difference between ideal simplicial volume and the ordinary
simplicial volume of a manifold M is that this new invariant measures the minimal size of possibly ideal triangulations of M "with real coefficients", since ideal simplices are now allowed to appear in representatives of the fundamental class.

Moreover, we will show that for manifolds whose boundary components all have an amenable funda-
mental group, the ideal simplicial volume coincides with the classical one.

Finally, if we have enough time, we will discuss the precise computation of the ideal simplicial volume
of an infinite family of hyperbolic 3-manifolds with geodesic boundary, for which the exact value of the
classical simplicial volume is not known.
This is a joint work with Roberto Frigerio