Geometriae Dedicata最新文献

Cup products provide a natural approach to access higher bounded cohomology groups. We extend vanishing results on cup products of Brooks quasimorphisms of free groups to cup products of median quasimorphisms, i.e., Brooks-type quasimorphisms of group actions on ({{,mathrm{{CAT}},}}(0)) cube complexes. In particular, we obtain such vanishing results for groups acting on trees and for right-angled Artin groups. Moreover, we outline potential applications of vanishing results for cup products in bounded cohomology.

Abstract

We study the group of all interval exchange transformations (IETs). Katok asked whether it contains a free subgroup. We show that for every IET S, there exists a dense open set (Omega (S)) of admissible IETs such that the group generated by S and any (Tin Omega (S)) is not free of rank 2. This extends a result by Dahmani et al. (Groups Geom Dyn 7(4):883–910, 2013): the group generated by a generic pair of elements of IET([0;1)) is not free (assuming a suitable condition on the underlying permutation).

We prove a conjecture of Ian Agol: all isometric realizations of a polyhedral surface with boundary sweep out an isotropic subset in the Kapovich-Millson moduli space of polygons isomorphic to the boundary. For a generic polyhedral disk we show that boundaries of its isometric realizations make up a Lagrangian subset. As an application of this result, we conclude that a generic equilateral polygon cannot be domed (in the sense of a problem of Kenyon, Glazyrin and Pak).

In this article, we consider the semipositive (resp. nef) line bundle on compact Kähler parabolic (resp. hyperbolic) manifolds. We prove some vanishing theorems for the (L^{2})-harmonic (n, q)-form of the holomorphic line bundles over complete Kähler manifolds.

For a pseudo-Anosov homeomorphism f on a closed surface of genus (gge 2), for which the entropy is on the order (frac{1}{g}) (the lowest possible order), Farb-Leininger-Margalit showed that the volume of the mapping torus is bounded, independent of g. We show that the analogous result fails for a surface of fixed genus g with n punctures, by constructing pseudo-Anosov homeomorphism with entropy of the minimal order (frac{log n}{n}), and volume tending to infinity.

We study metrics on two-dimensional simplicial complexes that are conformal either to flat Euclidean metrics or to the ideal hyperbolic metrics described by Charitos and Papadopoulos. Extending the results of our previous paper, we prove existence, uniqueness, and regularity results for harmonic maps between two such metrics on a complex.

Complex projective algebraic varieties with ({{mathbb {C}}}^*)-actions can be thought of as geometric counterparts of birational transformations. In this paper we describe geometrically the birational transformations associated to rational homogeneous varieties endowed with a ({{mathbb {C}}}^*)-action with no proper isotropy subgroups.

We study the space of Hopf differentials of almost fuchsian minimal immersions of compact Riemann surfaces. We show that the extrinsic curvature of the immersion at any given point is a concave function of the Hopf differential. As a consequence, we show that the set of all such Hopf differentials is a convex subset of the space of holomorphic quadratic differentials of the surface. In addition, we address the non-equivariant case, and obtain lower and upper bounds for the size of this set.

We characterise the varieties appearing in the third row of the Freudenthal–Tits magic square over an arbitrary field, in both the split and non-split version, as originally presented by Jacques Tits in his Habilitation thesis. In particular, we characterise the variety related to the 56-dimensional module of a Chevalley group of exceptional type (mathsf {E_7}) over an arbitrary field. We use an elementary axiom system which is the natural continuation of the one characterising the varieties of the second row of the magic square. We provide an explicit common construction of all characterised varieties as the quadratic Zariski closure of the image of a newly defined affine dual polar Veronese map. We also provide a construction of each of these varieties as the common null set of quadratic forms.