The action of mapping class groups on de Rham quasimorphisms

We study the action of the mapping class group on the subspace of de Rham classes in the degree-two bounded cohomology of a hyperbolic surface. In particular, we show that the only fixed nontrivial finite-dimensional subspace is the one generated by the Euler class. As a consequence, we get that the action of the mapping class group on the space of de Rham quasimorphisms has no fixed points.Comment: 21 pages, 4 figures. Comments are welcome

Simplicial volume of open manifolds

The simplicial volume is a proper homotopy invariant of manifolds introduced by Gromov in the 80's. In the first part of the thesis, I prove that the only open contractible 3-manifold with vanishing simplicial volume is the Euclidean space, and that any other open contractible 3-manifold has infinite simplicial volume. With the same techniques, I compute the spectrum of simplicial volume of open irreducible 3-manifolds. This work is contained in a paper joint with Prof. Roberto Frigerio. In the second part, I prove that (under some technical hypotheses) an open manifold with amenable fundamental group at infinity has finite simplicial volume. I prove that the same conclusion holds for manifolds which are simply connected at infinity

The spectrum of simplicial volume

Nella tesi, ripercorriamo un articolo di Heuer e Loeh del 2019 ("The spectrum of simplicial volume"), in cui si dimostra che in dimensione maggiore o uguale a 4, i volumi simpliciali delle varietà connesse chiuse orientate di dimensione fissata formano un insieme denso nei numeri reali positivi. Inoltre, si riesce a dimostrare che in dimensione 4 ogni numero razionale positivo è realizzabile come volume simpliciale di una 4-varietà connessa chiusa orientata. La dimostrazione utilizza risultati di teoria dei gruppi per collegare la stable commutator length con la norma l^1 di alcune 2-classi nell'omologia singolare di certi gruppi; queste classi vengono promosse a classi di dimensione più alta usando il prodotto cross. Si utilizza infine un teorema dovuto a Thom per ricavare varietà con volume simpliciale controllato. In the thesis, we cover an article by Heuer and Loeh of 2019 ("The spectrum of simplicial volume"), where it is shown that in dimension greater than or equal to 4, the simplicial volumes of oriented closed connected manifolds of fixed dimension form a dense subset of the positive real half-line. Moreover, it is proved that in dimension 4 every positive rational number can be realized as simplicial volume of an oriented closed connected 4-manifold. The proof relies on group theoretic results to relate stable commutator length to l^1-norm of some 2-classes in singular homology of certain groups; these classes are promoted into classes of higher dimension using cross product. Eventually, thanks to a theorem by Thom we can obtain manifolds with controlled simplicial volume