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Gabi Ben Simon

Publications and source records attributed to Gabi Ben Simon.

10 recordsLinked to original sources
arXiv2016

On weakly maximal representations of surface groups

We introduce and study a new class of representations of surface groups into Lie groups of Hermitian type, called {\em weakly maximal} representations. We prove that weakly maximal representations are discrete and injective and we describe the structure of the Zariski closure of their image. Furthermore we prove that the set of weakly maximal representations is a closed subset of the representation variety and describe its relation to other geometrically significant subsets of the representation variety.

math.DG↗
arXiv2016

On order preserving representations

In this article we introduce order preserving representations of fundamental groups of surfaces into Lie groups with bi-invariant orders. By relating order preserving representations to weakly maximal representations, introduced in arXiv:1305.2620, we show that order preserving representations into Lie groups of Hermitian type are faithful with discrete image and that the set of order preserving representations is closed in the representation variety. For Lie groups of Hermitian type whose associated symmetric space is of tube type we give a geometric characterization of these representations in terms of the causal structure on the Shilov boundary.

math.DG↗
arXiv2012

A Family of Quasimorphism Constructions

In this work we present a principle which says that quasimorphisms can be obtained via "local data" of the group action on certain appropriate spaces. In a rough manner the principle says that instead of starting with a given group and try to build or study its space of quasimorphisms, we should start with a space with a certain structure, in such a way that groups acting on this space and respect this structure will automatically carry quasimorphisms, where these are suppose to be better understood. In this paper we suggest such a family of spaces and give demonstrating examples for countable groups, groups that relate to action on the circle as well as outline construction for diffeomorphism groups. A distinctive advantage of this principle is that it allows the construction of the quasimorphism in a quite direct way. Further, we prove a lemma which besides serving as a platform for the construction of quasimorphisms on countable groups, bare interest by itself. Since it provides us with an embedding of any given countable group as a group of quasi-isometries of a universal space, where this space of embeddings is in bijection with the projective space of the homogeneous quasimorphism space of the group.

math.GR↗
arXiv2011

Weakly maximal representations of surface groups

We introduce and study a new class of representations of surface groups into Lie groups of Hermitian type, called weakly maximal representations. They are defined in terms of invariants in bounded cohomology and extend considerably the scope of maximal representations. We prove that weakly maximal representations are discrete and injective and describe the structure of the Zariski closure of the image. An interesting feature of these representations is that they admit an elementary topological characterization in terms of bi-invariant orderings. In particular if the target group is Hermitian of tube type, the ordering can be described in terms of the causal structure on the Shilov boundary.

math.GR↗
arXiv2011

Invariant Orders on Hermitian Lie Groups

We study three natural bi-invariant partial orders on a certain covering group of the automorphism group of a bounded symmetric domain of tube type; these orderings are defined using the geometry of the Shilov boundary, Lie semigroup theory and quasimorphisms respectively. Our main result shows that these orders are related by two inclusion relations. In the case of $SL_2(\R)$ we can show that they coincide.

math.GR↗
arXiv2011

Quasi-Total Orders and Translation Numbers

We show that a group admits a non-zero homogeneous quasimorphism if and only if it admits a certain type of action on a poset. Our proof is based on a construction of quasimorphisms which generalizes Poincaré--Ghys' construction of the classical translation number quasimorphism. We then develop a correspondence between quasimorphisms and actions on posets, which allows us to translate properties of orders into properties of quasimorphisms and vice versa. Concerning examples we obtain new realizations of the Rademacher quasimorphism, certain Brooks type quasimorphisms, the Dehornoy floor quasimorphism as well as Guichardet-Wigner quasimorphisms on simple Hermitian Lie groups of tube type. The latter we relate to Kaneyuki causal structures on Shilov boundaries, following an idea by Clerc and Koufany. As applications we characterize those quasimorphisms which arise from circle actions, and subgroups of Hermitian Lie groups with vanishing Guichardet-Wigner quasimorphisms.

math.GR↗
arXiv2010

Reconstructing quasimorphisms from associated partial orders and a question of Polterovich

We show that every continuous homogeneous quasimorphism on a finite-dimensional 1-connected simple Lie group arises as the relative growth of any continuous bi-invariant partial order on that group. More generally we show, that an arbitrary homogeneous quasimorphism can be reconstructed as the relative growth of a partial order subject to a certain sandwich condition. This provides a link between invariant orders and bounded cohomology and allows the concrete computation of relative growth for finite dimensional simple Lie groups as well as certain infinite-dimensional Lie groups arising from symplectic geometry.

math.GR↗
arXiv2006

The geometry of partial order on contact transformations of prequantization manifolds

In this paper we find connection between the Hofer's metric of the group of Hamiltonian diffeomorphisms of a closed symplectic manifold, with an integral symplectic form, and the geometry, defined in a paper by Eliashberg and Polterovich, of the quantomorphisms group of its prequantization manifold. This gives two main results: First, we calculate, partly, the geometry of the quantomorphisms groups of a prequantization manifolds of an integral symplectic manifold which admits certain Lagrangian foliation. Second, for every prequantization manifold we give a formula for the distance between a point and a distinguished curve in the metric space associated to its group of quantomorphisms. Moreover, our first result is a full computation of the geometry related to the symplectic linear group which can be considered as a subgroup of the contactomorphisms group of suitable prequantization manifolds of the complex projective space. In the course of the proof we use in an essential way the Maslov quasimorphism.

math.SG↗
arXiv2006

The Nonlinear Maslov index and the Calabi homomorphism

In this paper we find that the asymptotic nonlinear Maslov index defined on the universal cover of the group of all contact Hamiltonian diffeomorphisms of the standard 2n-1 dimensional contact sphere is a quasimorphism. Then we show our main result: Let M be standard n-1 complex projective space. We prove that the value of the pull back of the asymptotic nonlinear Maslov index to the universal cover of the group of Hamiltonian diffeomorphisms of M, restricted to elements supported in sufficiently small open subsets of M, equals the Calabi invariant of these elements up to a multiplication by a constant.

math.SG↗