This is the second part of a series of three articles which introduce laminations for free groups (see math.GR/0609416 for the first part). Several definition of the dual lamination of a very small action of a free group on an $\R$-tree are given and proved to be equivalent.

$\R$-trees and laminations for free groups II: The dual lamination of an $\R$-tree

We prove that every simplicial automorphism of the free splitting graph of a free group Fn is induced by an outer automorphism of Fn for n ≥ 3. In this note we consider the graph Gn of free splittings of the free group Fn of rank n ≥ 3. Loosely speaking, Gn is the graph whose vertices are non-trivial free splittings of Fn up to conjugacy, and where two vertices are adjacent if they are represented by free splittings admitting a common refinement. The group Out(Fn) of outer automorphisms of Fn acts simplicially on Gn. Denoting by Aut(Gn) the group of simplicial automorphisms of the free splitting graph, we prove: Theorem 1. The natural map Out(Fn)→ Aut(Gn) is an isomorphism for n ≥ 3. We briefly sketch the proof of Theorem 1. We identify Gn with the 1skeleton of the sphere complex Sn and observe that every automorphism of Gn extends uniquely to an automorphism of Sn. It is due to Hatcher [4] that the sphere complex contains an embedded copy of the spine Kn of Culler-Vogtmann space. We prove that the latter is invariant under Aut(Sn), and that the restriction homomorphism Aut(Sn) → Aut(Kn) is injective. The claim of Theorem 1 then follows from a result of Bridson-Vogtmann [1] which asserts that Out(Fn) is the full automorphism group of Kn. Before concluding this introduction we would like to point out that recently Martino and Francaviglia [9] have proved that Out(Fn) is also the full isometry group of Culler-Vogtmann space when the latter is endowed with the Lipschitz metric. While one could claim that Theorem 1 is the analogue of Ivanov’s theorem on the isometries of the curve complex [6], the result of Martino and Francaviglia is the analogue of Royden’s theorem on the isometries of Teichmuller space [10]. The first author was partially supported by M.E.C. grant MTM2006/14688 and NUI Galway’s Millennium Research Fund; the second author was partially supported by NSF grant DMS-0706878 and Alfred P. Sloan Foundation.

/pdf/automorphisms-of-the-graph-of-free-splittings-5c49skat90.pdf

Automorphisms of the graph of free splittings

Handel and Mosher have proved that the free splitting complex FS for the free group is Gromov hyperbolic. This is a deep and much sought-after result, since it establishes FS as a good analogue of the curve complex for surfaces. We give a shorter alternative proof of this theorem, using surgery paths in Hatcher's sphere complex (another model for the free splitting complex), instead of Handel and Mosher's fold paths. As a byproduct, we get that surgery paths are unparameterized quasi-geodesics in the sphere complex.

The hyperbolicity of the sphere complex via surgery paths

We prove that every simplicial automorphism of the free splitting graph of a free group of at least rank 3 is induced by an outer automorphism of the free group.

We prove that if a normal subgroup of the extended mapping class group of a closed surface has an element of sufficiently small support then its automorphism group and abstract commensurator group are both isomorphic to the extended mapping class group. The proof relies on another theorem we prove, which states that many simplicial complexes associated to a closed surface have automorphism group isomorphic to the extended mapping class group. These results resolve the metaconjecture of N.V. Ivanov, which asserts that any "sufficiently rich" object associated to a surface has automorphism group isomorphic to the extended mapping class group, for a broad class of such objects. As applications, we show: (1) right-angled Artin groups and surface groups cannot be isomorphic to normal subgroups of mapping class groups containing elements of small support, (2) normal subgroups of distinct mapping class groups cannot be isomorphic if they both have elements of small support, and (3) distinct normal subgroups of the mapping class group with elements of small support are not isomorphic. Our results also suggest a new framework for the classification of normal subgroups of the mapping class group.

Normal subgroups of mapping class groups and the metaconjecture of Ivanov.

The purpose of this article is to study co-dimension $2$ iso-contact embeddings of closed contact manifolds. We first show that a closed contact manifold $(M^{2n-1}, \xi_M)$ iso-contact embeds in a contact manifold $(N^{2n+1}, \xi_N),$ provided $M$ contact embeds in $(N, \xi_N)$ with a trivial normal bundle and the contact structure induced on $M$ via this embedding is homotopic as an almost-contact structure to $\xi_M.$ We apply this result to first establish that a closed contact $3$--manifold having no $2$--torsion in its second integral cohomology iso-contact embeds in the standard contact $5$--sphere if and only if the first Chern class of the contact structure is zero. Finally, we discuss iso-contact embeddings of closed simply connected contact $5$--manifolds.

Iso-contact embeddings of manifolds in co-dimension $2$

http://math.iisc.ernet.in/~gadgil/papers/IntSphr.pdf

Algebraic and geometric intersection numbers for free groups

In this note, we discuss embeddings of $3$--manifolds via open books. First we show that every open book of every closed orientable $3$--manifold admits an open book embedding in any open book decompistion of $S^2 \times S^3$ and $S^2 \widetilde{\times} S^3$ with the page a disk bundle over $S^2$ and monodromy the identity. We then use open book embeddings to reprove that every closed orientable $3$--manifold embeds in $S^5.$

Embeddings of $3$--manifolds via open books

For n >2, we shall show that the group Aut(NS(M)) of simplicial automorphisms of the complex NS(M) of non-separating embedded spheres in the manifold M,connected sum of n copies of S^2 X S^1, isomorphic to the group Out(F_n) of outer automorphisms of the free group F_n, where $F_n$ is identified with the fundamental group of M up to conjugacy of the base point in M.

https://projecteuclid.org/download/pdf_1/euclid.rmjm/1422885106

The complex of non-separating embedded spheres

In this note, we discuss open book embeddings of closed non-orientable $3$-manifolds in $5$-manifolds. We also show that a huge class of closed orientable $3$-manifolds, namely the orientation double covers of certain closed non-orientable $3$-manifolds, open book embeds in the $5$-sphere $S^5$. Finally, we give a new proof of a well-known theorem which states that every closed non-orientable $3$-manifold smoothly embeds in $S^5$.

Suhas Pandit

Papers

Iso-contact embeddings of manifolds in co-dimension $2$

Algebraic and geometric intersection numbers for free groups

Embeddings of $3$--manifolds via open books

The complex of non-separating embedded spheres

Open book embeddings of closed non-orientable $3$-manifolds