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Register a new userBounded geometry with no bounded pants decomposition
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We construct a quasiconformally homogeneous hyperbolic Riemann surface-other than the hyperbolic plane-that does not admit a bounded pants decomposition. Also, given a connected orientable topological surface of infinite type with compact boundary components, we construct a complete hyperbolic metric on the surface that has bounded geometry but does not admit a bounded pants decomposition.
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Using tropical geometry, Mikhalkin has proved that every smooth complex hypersurface in $mathbb{CP}^{n+1}$ decomposes into pairs of pants: a pair of pants is a real compact $2n$-manifold with cornered boundary obtained by removing an open regular neighborhood of $n+2$ generic hyperplanes from $mathbb{CP}^n$. As is well-known, every compact surface of genus $ggeqslant 2$ decomposes into pairs of pants, and it is now natural to investigate this construction in dimension 4. Which smooth closed 4-manifolds decompose into pairs of pants? We address this problem here and construct many examples: we prove in particular that every finitely presented group is the fundamental group of a 4-manifold that decomposes into pairs of pants.
This work deals with relations between a bounded cohomological invariant and the geometry of Hermitian symmetric spaces of noncompact type. The invariant, obtained from the Kahler class, is used to define and characterize a special class of totally geodesic embeddings, called tight embeddings. In addition, special isometric actions of finitely generated groups on Hermitian symmetric spaces are studied. Results of a joint work with M. Burger and A. Iozzi about surface group representations are also discussed.
We study collapsed manifolds with Ricci bounded covering geometry i.e., Ricci curvature is bounded below and the Riemannian universal cover is non-collapsed or consists of uniform Reifenberg points. Via Ricci flows techniques, we partially extend the nilpotent structural results of Cheeger-Fukaya-Gromov, on collapsed manifolds with (sectional curvature) local bounded covering geometry, to manifolds with (global) Ricci boundedcovering geometry.
We will prove the relative homotopy principle for smooth maps with singularities of a given {cal K}-invariant class with a mild condition. We next study a filtration of the group of homotopy self-equivalences of a given manifold P by considering singularities of non-negative {cal K}-codimensions.
This paper is devoted to the construction of norm-preserving maps between bounded cohomology groups. For a graph of groups with amenable edge groups we construct an isometric embedding of the direct sum of the bounded cohomology of the vertex groups in the bounded cohomology of the fundamental group of the graph of groups. With a similar technique we prove that if (X,Y) is a pair of CW-complexes and the fundamental group of each connected component of Y is amenable, the isomorphism between the relative bounded cohomology of (X,Y) and the bounded cohomology of X in degree at least 2 is isometric. As an application we provide easy and self-contained proofs of Gromov Equivalence Theorem and of the additivity of the simplicial volume with respect to gluings along pi_1-injective boundary components with amenable fundamental group.
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