Martin R. Bridson, Daniel P. Groves
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.
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Quadratic Isoperimetric Inequality for Mapping Tori of Hyperbolic Groups
该研究确立了一端无挠双曲群的映射环面具有二次等周不等式，推进了几何群论相关问题的研究。
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.
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 $ϕ$.
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.
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.
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。
We study the automorphisms ϕof a finitely generated free group F. Building on the train-track technology of Bestvina, Feighn and Handel, we provide a topological representative f:G\to G of a power of ϕthat behaves very much like the realization on the rose of a positive automorphism. This resemblance is encapsulated in the Beaded Decomposition Theorem which describes the structure of paths in G obtained by repeatedly passing to f-images of an edge and taking subpaths. This decomposition is the key to adapting our proof of the quadratic isoperimetric inequality for $F\rtimes_ϕ\mathbb Z$, with ϕ…
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。
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.
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。
We show the existence of several new infinite families of polynomially-growing automorphisms of free groups whose mapping tori are CAT(0) free-by-cyclic groups. Such mapping tori are thick, and thus not relatively hyperbolic. These are the first families comprising infinitely many examples for each rank of the nonabelian free group; they contrast strongly with Gersten's example of a thick free-by-cyclic group which cannot be a subgroup of a CAT(0) group.
The Conjugacy Problem for $Out(F_3)$
本文利用可计算不变量、train track 与映射环面，证明 $Out(F_3)$ 中两类剩余情形的共轭问题可判定，从而解决 $Out(F_3)$ 的共轭问题。
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…
In this note we study the problem of characterizing the complex affine space $\mathbb{A}^n$ via its automorphism group. We prove the following. Let $X$ be an irreducible quasi-projective $n$-dimensional variety such that $\mathrm{Aut}(X)$ and $\mathrm{Aut}(\mathbb{A}^n)$ are isomorphic as abstract groups. If $X$ is either quasi-affine and toric or $X$ is smooth with Euler characteristic $χ(X) \neq 0$ and finite Picard group $\mathrm{Pic}(X)$, then $X$ is isomorphic to $\mathbb{A}^n$. The main ingredient is the following result. Let $X$ be a smooth irreducible quasi-projective variety of dimens…
Dynamics of Dehn Twists in the Outer Automorphism Group of a Free Group
研究有限生成非交换自由群外自同构群中的Dehn扭转，证明在兼容条件下，有限多个Dehn扭转的足够大幂次生成右角Artin群，并建立bigon-bihedron准则和交换条件。
We prove that any $\mathbb{Z}^p$-action ${\bf A}$ that acts by automorphisms of $\mathbb{Z}^q$ with a non-zero fixed-point set induces a unipotent factor of the $\mathbb{Z}^p$-action ${\bf A}$ which determines whether the action ${\bf A}$ is {\it liberated affine}, i.e. ${\bf A}$ is the linear part of a free affine $\mathbb{Z}^p$-action on the torus $T^q$. In general, it is not true that all unipotent $\mathbb{Z}^p$-actions $\bf{U}$ on $\mathbb{Z}^q$ are liberated affine: counter-examples appear for $q\geq 4$. But if the dimension of the fixed-point set of ${\bf U}$ regarded as a subspace of $…
Mapping class groups of exotic tori and actions by ${\rm SL}_d(\mathbb Z)$
该研究确定了维数非4的奇异环面中保向微分同痕类群到SL_d(Z)的同态为分裂满射的情形，计算了一类奇异环面的映射类群，证明了其SL_d(Z)作用的同调性质，并得出许多奇异环面无非平凡SL_d(Z)微分作用的结论。
A one-by-one exhaustion is a combinatorial/geometric condition which excludes eigenvalues from the spectra of Laplace and Schrödinger operators on graphs. Isoperimetric inequalities in graphs with a cocompact automorphism group provide an upper bound on the von Neumann dimension of the space of eigenfunctions. Any finitely generated indicable amenable group has a Cayley graph without eigenvalues. There exists a finitely generated group G with finite generating sets S and S' such that the adjacency operator of the Cayley graph of (G,S) has no eigenvalue while the adjacency operator of the Cayle…
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