Allen Hatcher, Karen Vogtmann, Nathalie Wahl
We correct the proof of Theorem 5 of the paper Homology stability for outer automorphism groups of free groups, by the first two authors.
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Let $\operatorname{Aut}(H_1(N_g;\mathbb Z_2),\cdot )$ be the group of automorphisms on the first homology group with $\mathbb Z_2$ coefficient of a closed non-orientable surface $N_g$ preserving the mod $2$ intersection form. In this paper, we obtain a finite presentation for $\operatorname{Aut}(H_1(N_g;\mathbb Z_2),\cdot )$. As applications we calculate the first homology group and the second homology group of $\operatorname{Aut}(H_1(N_g;\mathbb Z_2),\cdot )$.
The main goal of this paper is a calculation of the integral (co)homology of the group of symmetric automorphisms of a free product. We proceed by giving a geometric interpretation of symmetric automorphisms via a moduli space of certain diagrams, which we name cactus products. To describe this moduli space a theory of diagonal complexes is introduced. This offers a generalisation of the theory of right-angled Artin groups in that each diagonal complex defines what we call a diagonal right-angled Artin group (DRAAG).
We define an integer-valued invariant of special cube complexes called the genus, and prove that having genus one characterizes special cube complexes with abelian fundamental group. Using the genus, we obtain a new proof that the fundamental group of a special cube complex is either free abelian or surjects onto a non-cyclic free group. We also investigate automorphisms of special cube complexes, and give a new geometric proof that the Torelli subgroup for a right-angled Artin group is torsion-free.
We study the word length entropy of automorphisms of residually nilpotent groups, and how the entropy of such group automorphisms relates to the entropy of induced automorphisms on various nilpotent quotients. We show that much like the structure of a nilpotent group is dictated to a large degree by its abelianization, the entropy of an automorphism of a nilpotent group is dictated by its entropy on the abelianization. We give some applications to the study of pseudo-Anosov homeomorphisms of surfaces. In particular, we show that if $ψ$ is a non--homological pseudo-Anosov homeomorphism of a sur…
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 SAut(F_n) admits no non-trivial actions by homeomorphisms on acyclic manifolds or spheres of smaller dimension. Indeed, SAut(F_n) cannot act non-trivially on any generalized Z_2-homology sphere of dimension less than n-1, nor on any Z_2-acyclic Z_2-homology manifold of dimension less than n. It follows that SL(n,Z) cannot act non-trivially on such spaces eithe…
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 groups. In particular, Aut(F_n) has no stable rational homology, settling a conjecture of Hatcher and Vogtmann.
We prove that if F is a finitely generated free group and f:F -> F is an automorphism with polynomial growth of degree d, then there exists a characteristic subgroup S < F of finite index such that the induced automorphism of the abelianisation of S also grows polynomially of degree d. The proof is geometric in nature and makes use of improved relative train track representatives.
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。
We compute the homology groups $H_*(\operatorname{Out}(F_7);\mathbb Q)$ of the outer automorphism group of the free group of rank $7$. We produce in this manner the first rational homology classes of $\operatorname{Out}(F_n)$ that are neither constant ($*=0$) nor Morita classes ($*=2n-4$).
We compute the rational stable homology of the automorphism groups of free nilpotent groups. These groups interpolate between the general linear groups over the ring of integers and the automorphism groups of free groups, and we employ functor homology to reduce to the abelian case. As an application, we also compute the rational stable homology of the outer automorphism groups and of the mapping class groups of the associated aspherical nil-manifolds in the TOP, PL, and DIFF categories.
The stable Albanese homology of the IA-automorphism groups of free groups
本文研究了自由群IA-自同构群的Albanese同调，确定了其在n≥3时的表示结构，并揭示了其与外自同构群及Aut(Fn)稳定上同调的关系。
Trees, homology, and automorphism groups of RAAGs
本文针对定义图为树的 RAAG，研究其自同构群中显式有限指数子群的同调，利用深顶点的数量与度数给出第一 Betti 数下界，并组合分析其随树规模变化的平均值。
We study the cohomology of Aut(F_n) and Out(F_n) with coefficients in the modules \wedge^q H, \wedge H^*, Sym^q H or Sym^q H^*, where H is the Out(F_n)-module obtained by abelianising the free group F_n. For reasons which are not conceptually clear, taking coefficients in H and its related modules behaves in a far less trivial way than taking coefficients in H^* and its related modules. Based on a conjectural homology stability theorem for spaces of graphs labeled by a simply connected background space, we give a stable integral calculation of these groups in low degrees, and modulo a further …
The concept of an adapted homology basis for a prime order conformal automorphism of a compact Riemann surface extends to arbitrary finite groups of conformal automorphisms. Here we compute some examples of adapted homology bases for some groups of automorphisms. The method is to begin by apply the Schreier-Reidemeister rewriting process along with the Schreier-Reidemeister Theorem and then to eliminate generators and relations until there is one single large defining relation for the fundamental group in which every generator and its inverse occurs. We are then able to compute the action of t…
We show in this article that, for any group $G$ indecomposable for the free product * and non-isomorphic to $\mathbf{Z}$, the canonical inclusion ${\rm Aut}(G^{*n})\to {\rm Aut}(G^{* n+1})$ induces an isomorphism between the homology groups $H_i$ for $n\geq 2i+2$, as was conjectured by Hatcher and Wahl. In fact we show a little more --- in particular, the result is true for any group $G$ if we replace the automorphism group of the free product by the subgroup of symmetric automorphisms. For this purpose we use constructions and acyclicity results due to McCullough-Miller and Chen-Glover-Jensen…
Homological stability fails for the Cremona groups, yet their stable homology is non-trivial
本文研究克雷莫纳群，发现其同调稳定性不成立，但证明其稳定同调由非平凡无限循环空间表示，为这类群的同调研究提供了新视角。
The observation that a graph of rank $n$ can be assembled from graphs of smaller rank $k$ with $s$ leaves by pairing the leaves together leads to a process for assembling homology classes for $Out(F_n)$ and $Aut(F_n)$ from classes for groups $Γ_{k,s}$, where the $Γ_{k,s}$ generalize $Out(F_k)=Γ_{k,0}$ and $Aut(F_k)=Γ_{k,1}$. The symmetric group $Σ_s$ acts on $H_*(Γ_{k,s})$ by permuting leaves, and for trivial rational coefficients we compute the $Σ_s$-module structure on $H_*(Γ_{k,s})$ completely for $k \leq 2$. Assembling these classes then produces all the known nontrivial rational homology …
Scissors automorphism groups I: Homological stability and K-theory
本文证明在温和条件下剪刀自同构群的同调独立于对象，可由剪刀同余 K-理论谱表达，提供群论解释与计算方法，并在经典情形及后续中得出同调计算与结构结果。
If $Q$ is a group acting as a group of automorphisms of another group $G$ (with finite orbits), denote by $C_*(G)^Q$ the subcomplex of $Q$-invariant chains in the bar complex $C_*(G)$. In this paper, we study the homology of the complex $C_*(G)^Q$ and compute several examples.
We introduce a homology theory for subspace arrangements, and use it to extract a new system of numerical invariants from the Bieri-Neumann-Strebel invariant of a group. We use these to characterize when the set of basis conjugating outer automorphisms (a.k.a. the pure symmetric outer automorphism group) of a right-angled Artin group is itself a right-angled Artin group.
Let S be a compact Riemann surfaces of genus g >= 2 and G a conformal automoprhism group of order n acting on S. In this paper we give the definition of an adapted generating set and an adapted basis for the first homology group of such a compact Riemann surface. This generating set and basis reflect the action of G in as simple manner as possible. This can be seen in the matrix of the action of G which we obtain. We prove the existence of such a generating set and basis for any conformal group acting on such a surface and find the matrix. This extends our earlier results on adapted bases and …
The free factor complex and the dualizing module for the automorphism group of a free group
本文回答了Hatcher-Vogtmann的问题，证明了对于n=5，自由因子复形的顶同调群并非Aut(F_n)的对偶化模。
We prove a representation stability result for the second homology groups of Torelli subgroups of mapping class groups and automorphism groups of free groups. This strengthens the results of Boldsen-Hauge Dollerup and Day-Putman. We also prove a new representation stability result for the homology of certain congruence subgroups, partially improving upon the work of Putman-Sam. These results follow from a general theorem on syzygies of certain modules with finite polynomial degree.
Let $S$ be a compact Riemann surface and $G$ a group of conformal automorphisms of $S$ with $S_0 = S/G$. $S$ is a finite regular branched cover of $S_0$. If $U$ denotes the unit disc, let $Γ$ and $Γ_0$ be the Fuchsian groups with $S = U/Γ$ and $S_0 = U/{Γ_0}$. There is a group homomorphism of $Γ_0$ onto $G$ with kernel $Γ$ and this is termed a surface kernel map. Two surface kernel maps are equivalent if they differ by an automorphism of $Γ_0$. In his 1971 paper Harvey showed that when $G$ is a cyclic group, there is a unique simplest representative for this equivalence class. His result has p…
Unipotent automorphisms of modular group algebras and the homology of the Lazard Lie algebra
本文证明指数为p且类至多p-2的有限p-群的模群代数单幂自同构与群自同构在关系模上作用一致，并给出U(H)的Lazard李代数刻画，计算了类2群的U(H)大小。
Let F_{n+m} be the free group of rank n+m, with generators x_1,...,x_{n+m}. An automorphism ϕof F_{n+m} is called partially symmetric if for each 1 \le i \le m, ϕ(x_i) is conjugate to x_j or x_j^{-1} for some 1 \le j \le m. Let Σ\Aut_n^m be the group of partially symmetric automorphisms. We prove that for any m \ge 0 the inclusion Σ\Aut_n^m \to Σ\Aut_{n+1}^m induces an isomorphism in rational homology for dimensions i satisfying n \ge (3(i+1)+m)/2, with a similar statement for the groups PΣ\Aut_n^m of pure partially symmetric automorphisms. We also prove that for any n \ge 0 the inclusion Σ\Au…
Let F_n be the free group of rank n, with generating set S=\{x_1,...,x_n\}. An automorphism ϕof F_n is called symmetric if for each 1\leq i\leq n, ϕ(x_i) is conjugate to x_j or x_j^{-1} for some 1\leq j\leq n. Let ΣAut(F_n) be the group of symmetric automorphisms. We prove that the inclusion ΣAut(F_n) \rightarrow ΣAut(F_{n+1}) induces an isomorphism in rational homology for n>(3i-1)/2.
Representation stability for the first homology of congruence subgroups
研究映射类群和自由群自同构群的同余子群的第一同调，以及Torelli群交换化的余不变模在有限域上的模表示序列，确定其合成因子和重数，并建立Church-Farb意义下的周期表征稳定性，应用于研究带level 2结构的曲线模空间上的平坦线丛。
A homology cylinder over a compact manifold is a homology cobordism between two copies of the manifold together with a boundary parametrization. We study abelian quotients of the homology cobordism group of homology cylinders. For homology cylinders over general surfaces, it was shown by Cha, Friedl and Kim that their homology cobordism groups have infinitely generated abelian quotient groups by using Reidemeister torsion invariants. In this paper, we first investigate their abelian quotients again by using another invariant called the Magnus representation. After that, we apply the machinery …
Morita introduced in 2008 a 1-cocycle on the group of homology cobordisms of surfaces with values in an infinite-dimensional vector space. His 1-cocycle contains all the "traces" of Johnson homomorphisms which he introduced fifteen years earlier in his study of the mapping class group. In this paper, we propose a new version of Morita's 1-cocycle based on a simple and explicit construction. Our 1-cocycle is proved to satisfy several fundamental properties, including a connection with the Magnus representation and the LMO homomorphism. As an application, we show that the rational abelianization…
We study the group $QV$, the self-maps of the infinite $2$-edge coloured binary tree which preserve the edge and colour relations at cofinitely many locations. We introduce related groups $QF$, $QT$, $\tilde{Q}T$, and $\tilde{Q}V$, prove that $QF$, $\tilde{Q}T$, and $\tilde{Q}V$ are of type $\mathrm{F}_\infty$, and calculate finite presentations for them. We calculate the normal subgroup structure and rational homology of all $5$ groups, the Bieri--Neumann--Strebel--Renz invariants of $QF$, and discuss the relationship of all $5$ groups with other generalisations of Thompson's groups.
By analyzing how the Borel regulator classes vanish on various groups related to $\mathrm{GL}(n,\mathrm{Z})$, we define three series of secondary characteristic classes for subgroups of automorphism groups of free groups. The first case is the $\mathrm{IA}$-automorphism groups and we show that our classes coincide with higher $\mathrm{FR}$ torsions due to Igusa. The second case is the mapping class groups and our classes also turn out to be his higher torsions which are non-zero multiples of the Mumford-Morita-Miller classes of even indices. Our construction gives new group cocycles for these …
Nielsen classes in outer automorphism groups of free groups
针对 $n\ge8$，研究自由群外自同构群中生成对的尼尔森类，构造了 $GL(n,\mathbb Z)$ 等的生成对族，推导关系模障碍，Evans例子验证其可区分稳定化后的尼尔森类。
Let $V_r(Σ)$ be the generalised Thompson group defined as the automorphism group of a valid, bounded, and complete Cantor algebra. We show that that for every $n>0$ there is a $k>n,$ such that there exists a $k$-dimensional $n$-connected simplicial complex $K$ such that $V_r(Σ)$ acts on $K$ with finite stabilisers. We also determine the number of conjugacy classes of finite cyclic subgroups of a given order $m$ in Brin-Thompson groups. We apply our computations to the rationalised Farrell-Jones assembly map in algebraic $K$-theory.
We calculate the homology of automorphism groups of free groups in various cases. The author would like to thank his thesis advisor Karen Vogtmann.
Let $n\ge 3$, and let $\mathrm{Out}(W_n)$ be the outer automorphism group of a free Coxeter group $W_n$ of rank $n$. We study the growth of the dimension of the homology groups (with coefficients in any field $\mathbb{K}$) along Farber sequences of finite-index subgroups of $\mathrm{Out}(W_n)$. We show that, in all degrees up to $\lfloor\frac{n}{2}\rfloor-1$, these Betti numbers grow sublinearly in the index of the subgroup. When $\mathbb{K}=\mathbb{Q}$, through Lück's approximation theorem, this implies that all $\ell^2$-Betti numbers of $\mathrm{Out}(W_n)$ vanish up to degree $\lfloor\frac{n…
Scissors automorphism groups II: Solomon-Tits theorems
本文证明了在欧几里得、双曲或球面几何中，由点或超平面生成的测地子空间集合的正则非平凡子空间的poset的实现在球面 wedge 中等价，为后续计算剪刀自同构群的同调奠定基础。
Let $X$ be a smooth manifold belonging to one of these three collections: acyclic manifolds (compact or not, possibly with boundary), compact connected manifolds (possibly with boundary) with nonzero Euler characteristic, integral homology spheres. We prove that $Diff(X)$ is Jordan. This means that there exists a constant $C$ such that any finite subgroup $G$ of $Diff(X)$ has an abelian subgroup whose index in $G$ is at most $C$. Using a result of Randall and Petrie we deduce that the automorphism groups of connected, non necessarily compact, smooth real affine varieties with nonzero Euler cha…
Infinitely many Nielsen classes in mapping class groups and free-group automorphism groups
本文提出根共轭准则，利用多项式对称迹不变量检测 Nielsen 不等价性，证明映射类群和自由群自同构群具有可数无穷多个生成对的 Nielsen 类，并回答了两个开放问题。
In the paper "Aquino, C., Jiménez, R., Mijangos, M., Morales Meléndez, Q.: On Invariant (co)homology of a group, preprint" are introduced two groups generated by the orbits of an action of a group on another group by automorphisms. One is of group-theoretic nature and the other comes from homology of invariant group chains. In this note are given some properties of the first groups and is studied a natural homomorphism between these groups. More precisely, it is shown that this homomorphism is not injective nor surjective. A description of the kernel is given. Note: there was a previous versio…
In view of a potential interpretation of the role of the Mathieu group M_24 in the context of strings compactified on K3 surfaces, we develop techniques to combine groups of symmetries from different K3 surfaces to larger 'overarching' symmetry groups. We construct a bijection between the full integral homology lattice of K3 and the Niemeier lattice of type (A_1)^24, which is simultaneously compatible with the finite symplectic automorphism groups of all Kummer surfaces lying on an appropriate path in moduli space connecting the square and the tetrahedral Kummer surfaces. The Niemeier lattice …
Frattini Geometry of Non-Generating Complexes: Symmetry, Homology and Betti Numbers
本文通过Frattini拓扑约化将有限群的非生成复形化为初等阿贝尔向量空间的非张成复形，从而确定其同构、自同构群、同调及Betti数，并给出完整Cohen-Macaulay分类。
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。
The walled Brauer category and stable cohomology of $\mathrm{IA}_n$
本文借助有墙 Brauer 范畴改造 Kupers–Randal-Williams 方法，在稳定有限维假设下刻画 IA-自同构群稳定有理上同调的代数部分至次数 Q+1，并应用于稳定 Albanese（共）同调。
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