Mark Feighn, Michael Handel
The mapping torus of an endomorphism Φof a group G is the HNN-extension G*_G with bonding maps the identity and Φ. We show that a mapping torus of an injective free group endomorphism has the property that its finitely generated subgroups are finitely presented and, moreover, these subgroups are of finite type.
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If $F$ is a finitely generated free group and $ϕ$ is a positive automorphism of $F$ then $F\rtimes_ϕZ$ satisfies a quadratic isoperimetric inequality.
We present necessary and sufficient conditions for the existence of a splitting over $\mathbb{Z}$ of the mapping torus $M_ϕ=F\rtimes \mathbb{Z}$ of a free group automorphism $ϕ$.
If F is a finitely generated free group and ϕis an automorphism of F then the mapping torus of ϕadmits a quadratic isoperimetric inequality. This is the third and final paper in a series proving this theorem. The first two were math.GR/0211459 and math.GR/0507589.
If F is a finitely generated free group and ϕis an automorphism of F then F \rtimes_ϕZ satisfties a quadratic isoperimetric inequality.
Let $G$ be the mapping torus of a polynomially growing automorphism of a finitely generated free group. We determine which epimorphisms from $G$ to $\mathbb{Z}$ have finitely generated kernel, and we compute the rank of the kernel. We thus describe all possible ways of expressing $G$ as the mapping torus of a free group automorphism. This is similar to the case for 3--manifold groups, and different from the case of mapping tori of exponentially growing free group automorphisms. The proof uses a hierarchical decomposition of $G$ and requires determining the Bieri-Neumann-Strebel invariant of th…
An n-dimensional quantum torus is a twisted group algebra of the group $\Z^n$. It is called rational if all invertible commutators are roots of unity. In the present note we describe a normal form for rational n-dimensional quantum tori over any field. Moreover, we show that for $n = 2$ the natural exact sequence describing the automorphism group of the quantum torus splits over any field.
For an amalgam of two free groups and a particular kind of automorphism, we show that the Dehn function of the corresponding mapping torus is quadratic.
We give a cohomological criterion for existence of outer automorphisms of a semisimple algebraic group over an arbitrary field. This criterion is then applied to the special case of groups of type D_2n over a global field, which completes some of the main results from the paper "Weakly commensurable arithmetic groups and isospectral locally symmetric spaces" (Pub. Math. IHES, 2009) by Prasad and Rapinchuk and gives a new proof of a result from another paper by the same authors.
Homology growth of polynomially growing mapping tori
本文研究多项式增长自同构的映射圆柱的同调增长性质，证明其具有廉价重建性质并显示torsion同调增长在Farber序列下消失。
Reducing the conjugacy problem for relatively hyperbolic automorphisms to peripheral components
本研究将自由群与无挠双曲群的外自同构共轭问题归约为多项式增长自同构映射环面的相关算法问题，求解了多类新外自同构的共轭问题，为彻底解决$Out(F_n)$的共轭问题提供了可行路径。
We study smooth actions of SAut(Fn), the unique subgroup of index two in the automorphism group of a free group of rank n, as a part of the generalized 'Zimmer program'. In particular, we show that every action of SAut(Fn) on a low dimensional torus is trivial.
Cubulating mapping tori of some polynomial growth free group automorphisms
本文给出了多项式增长自由群自同构映射环自由作用于 CAT(0) 立方复形的充分条件，利用循环层级与立方小消去理论证明随机幂单多项式增长自同构的映射环可被立方化。
Given an octonion algebra over a field k, its automorphism group G is an algebraic semisimple k-group of type G_2. We study the maximal tori of G in terms of the algebra C.
Answering a question of A. Rapinchuk, we construct examples of non- isomorphic semisimple algebraic groups H1 and H2 of type G2 with coherently equivalent systems of maximal k-tori.
The minimal degree of a permutation group $G$ is the minimum number of points not fixed by non-identity elements of $G$. Lower bounds on the minimal degree have strong structural consequences on $G$. Babai conjectured that if a primitive coherent configuration with $n$ vertices is not a Cameron scheme, then its automorphism group has minimal degree $\geq cn$ for some constant $c>0$. In 2014, Babai proved the desired lower bound on the minimal degree of the automorphism groups of strongly regular graphs, thus confirming the conjecture for primitive coherent configurations of rank 3. In this pap…
We present an algorithm which, given any finite presentation of a group as input, will terminate with answer yes if and only if the group is large. We use this to prove that a mapping torus of a finitely generated free group automorphism is large if it contains the integers times the integers as a subgroup of infinite index. We then extend this result to mapping tori of finitely generated free group endomorphisms, as well as showing that such a group is large if it contains a Baumslag-Solitar group of infinite index and has a finite index subgroup with first Betti number at least 2. We also sh…
Dehn functions of mapping tori of right-angled Artin groups
本文研究直角 Artin 群在自同构下的代数映射环面，并对所有 3 生成元直角 Artin 群及 F_k × F_l（k,l ≥ 2）按自同构 Φ 分类其 Dehn 函数。
It has been proven by Schupp and Bergman that the inner automorphisms of groups can be characterized purely categorically as those group automorphisms that can be coherently extended along any outgoing homomorphism. One is thus motivated to define a notion of (categorical) inner automorphism in an arbitrary category, as an automorphism that can be coherently extended along any outgoing morphism, and the theory of such automorphisms forms part of the theory of covariant isotropy. In this paper, we prove that the categorical inner automorphisms in any category $\mathsf{Group}^{\mathcal{J}}$ of p…
In this article we show how to calculate the group of automorphisms of flat Kähler manifolds. Moreover we are interested in the problem of classification of such manifolds up to biholomorphism. We consider these problems from two points of view. The first one treats the automorphism group as a subgroup of the group of affine transformations, while in the second one we analyze it using automorphisms of complex tori. This leads us to the analogues of the Bieberbach theorems in the complex case. We end with some examples, which in particular show that in general the finiteness of the automorphism…
Coherent configurations (CCs) are highly regular colorings of the set of ordered pairs of a "vertex set"; each color represents a "constituent digraph." CCs arise in the study of permutation groups, combinatorial structures such as partially balanced designs, and the analysis of algorithms; their history goes back to Schur in the 1930s. A CC is primitive (PCC) if all its constituent digraphs are connected. We address the problem of classifying PCCs with large automorphism groups. This project was started in Babai's 1981 paper in which he showed that only the trivial PCC admits more than $\exp(…
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