Related papers: The Complex of Non-separating embedded spheres

The outer automorphism group Out(F_2g) of a free group on 2g generators naturally contains the mapping class group of a punctured surface as a subgroup. We define a subsurface projection of the sphere complex of the connected sum of n…

Let S be any orientable surface of infinite genus with a finite number of boundary components. In this work we consider the curve complex C(S), the nonseparating curve complex N(S) and the Schmutz graph G(S) of S. When all the topological…

We prove that the simplicial complex whose simplices are the nonempty partial bases of $\mathbb{F}_n$ is homotopy equivalent to a wedge of $(n-1)$-spheres. Moreover, we show that it is Cohen-Macaulay.

Let $N$ be a connected nonorientable surface of genus $g$ with $n$ punctures. Suppose that $g$ is odd and $g+n \geqslant 6$. We prove that the automorphism group of the complex of curves of $N$ is isomorphic to the mapping class group…

The BNSR-invariants of a group $G$ are a sequence $\Sigma^1(G)\supseteq \Sigma^2(G) \supseteq \cdots$ of geometric invariants that reveal important information about finiteness properties of certain subgroups of $G$. We consider the…

The automorphism group of a finitely generated free group is the normal closure of a single element of order 2. If $m$ is less than $n$ then a homomorphism $Aut(F_n)\to Aut(F_m)$ can have cardinality at most 2. More generally, this is true…

For n at least 3, let SAut(F_n) denote the unique subgroup of index two in the automorphism group of a free group. The standard linear action of SL(n,Z) on R^n induces non-trivial actions of SAut(F_n) on R^n and on S^{n-1}. We prove that…

We demonstrate the existence of minimal simplicial $n$-complexes which inevitably contain a nonsplittable two-component link formed by an $(n-1)$-sphere and an $n$-sphere in any embedding into $\mathbb{R}^{2n}$. This provides a…

Let X be an irreducible symplectic manifold and L a nef line bundle on X which is isotropic with respect to the Beauville-Bogomolov quadratic form. It is known that a subgroup Aut(X,L) of an automorphism group of X which fix L is almost…

Let F_n be the free group of rank n, with generating set S=\{x_1,...,x_n\}. An automorphism \phi of F_n is called symmetric if for each 1\leq i\leq n, \phi(x_i) is conjugate to x_j or x_j^{-1} for some 1\leq j\leq n. Let \Sigma Aut(F_n) be…

Let $F_n= F\langle x_1,...,x_n\rangle$ denote the free group of rank $n\ge 2$ and let $\mathrm{End}(F_n)$ be the endomorphism monoid of $F_n$. We show that automorphisms of $F_n$ are detected via the $\mathrm{End}(F_n)$-action on the first…

We compute the automorphism group of the dual complex $\mathsf{T}_{d, n}$ of the boundary divisor in the Kontsevich moduli space $\overline{\mathcal{M}}_{0, n}(\mathbb{P}^r, d)$. When $d \geq 2$, we find that $\mathrm{Aut}(\mathsf{T}_{d,…

We prove that any isometry of the graph of cyclic splittings of a finitely generated free group $F_N$ of rank $N\ge 3$ is induced by an outer automorphism of $F_N$. The same statement also applies to the graphs of maximally-cyclic…

For positive integers k,n, we investigate the simplicial complex NM_k(n) of all graphs G on vertex set [n] such that every matching in G has size less than k. This complex (along with other associated cell complexes) is found to be homotopy…

This paper contains examples of closed aspherical manifolds obtained as a by-product of recent work by the author [arXiv:math.GR/0509490] on the relative strict hyperbolization of polyhedra. The following is proved. (I) Any closed…

Let $M^{r}$ be a connected orientable manifold with the Euler characteristic $\chi(M)\not \equiv 0\operatorname{mod}6$. Denote by $\mathrm{SAut}(F_{n})$ the unique subgroup of index two in the automorphism group of a free group. Then any…

Homology of the group Aut(F_n) of automorphisms of a free group on n generators is known to be independent of n in a certain stable range. Using tools from homotopy theory, we prove that in this range it agrees with homology of symmetric…

We consider actions of automorphism groups of free groups by semisimple isometries on complete CAT$(0)$ spaces. If $n\ge 4$ then each of the Nielsen generators of Aut$(F_n)$ has a fixed point. If $n=3$ then either each of the Nielsen…

The free splitting graph of a free group $F_n$ with $n\geq 2$ generators is a hyperbolic ${\rm Out}(F_n)$-graph which has a geometric realization as a sphere graph in the connected sum of $n$ copies of $S^1\times S^2$. We use this…

We prove that the automorphism group of a Fra\"iss\'e structure M equipped with a notion of stationary independence is universal for the class of automorphism groups of substructures of M. Furthermore, we show that this applies to certain…