Quasi-homomorphisms and stable lengths in mapping class groups

Median structures on asymptotic cones and homomorphisms into mapping class groups

The main goal of this paper is a detailed study of asymptotic cones of the mapping class groups. In particular, we prove that every asymptotic cone of a mapping class group has a bi-Lipschitz equivariant embedding into a product of real trees, sending limits of hierarchy paths onto geodesics, and with image a median subspace. One of the applications is that a group with Kazhdan's property (T) can have only finitely many pairwise non-conjugate homomorphisms into a mapping class group. We also give a new proof of the rank conjecture of Brock and Farb (previously proved by Behrstock and Minsky, a…

纯映射类群之间的同态

Homomorphisms between pure mapping class groups

该研究证明了纯映射类群之间保持多扭的同态由多嵌入诱导，并对特定参数范围内的同态进行了分类。

Injective homomorphisms of mapping class groups of non-orientable surfaces

Let $N$ be a compact, connected, non-orientable surface of genus $ρ$ with $n$ boundary components, with $ρ\ge 5$ and $n \ge 0$, and let $\mathcal{M} (N)$ be the mapping class group of $N$. We show that, if $\mathcal{G}$ is a finite index subgroup of $\mathcal{M} (N)$ and $φ: \mathcal{G} \to \mathcal{M} (N)$ is an injective homomorphism, then there exists $f_0 \in \mathcal{M} (N)$ such that $φ(g) = f_0 g f_0^{-1}$ for all $g \in \mathcal{G}$. We deduce that the abstract commensurator of $\mathcal{M} (N)$ coincides with $\mathcal{M} (N)$.

Property $(TT)$ modulo $T$ and homomorphism superrigidity into mapping class groups

Every homomorphism from finite index subgroups of a universal lattices to mapping class groups of orientable surfaces (possibly with punctures), or to outer automorphism groups of finitely generated nonabelian free groups must have finite image. Here the universal lattice denotes the special linear group G=SL_m(Z[x1,...,xk]) with m at least 3 and k finite. Moreover, the same results hold ture if universal lattices are replaced with symplectic universal lattices Sp_{2m}(Z[x1,...,xk]) with m at least 2. These results can be regarded as a non-arithmetization of the theorems of Farb--Kaimanovich--…

值域递减群同态的一种表示

A representation of range decreasing group homomorphisms

研究整个映射群上的值域递减群同态，通过刻画该同态，确定某些映射群上此类同态并分类特殊同态，此前相关工作集中在映射群特定子集，此研究完善了对值域递减群同态的认知。

On Quasi-homomorphisms and Commutators in the Special Linear Group over a Euclidean Ring

We prove that for any euclidean ring R and n at least 6, Gamma=SL_n(R) has no unbounded quasi-homomorphisms. From Bavard's duality theorem, this means that the stable commutator length vanishes on Gamma. The result is particularly interesting for R = F[x] for a certain field F (such as the field C of complex numbers, because in this case the commutator length on Gamma is known to be unbounded. This answers a question of M. Abért and N. Monod for n at least 6.

Low-dimensional linear representations of mapping class groups

Recently, John Franks and Michael Handel proved that, for $g\geq 3$ and $n\leq 2g-4$, every homomorphism from the mapping class group of an orientable surface of genus $g$ to $\GL (n,\C)$ is trivial. We extend this result to $n\leq 2g-1$, also covering the case $g=2$. As an application, we prove the corresponding result for nonorientable surfaces. Another application is on the triviality of homomorphisms from the mapping class group of a closed surface of genus $g$ to $\Aut (F_n)$ or to $\Out (F_n)$ for $n\leq 2g-1$.

Simply Intersecting Pair Maps in the Mapping Class Group

The Torelli group, I(S_g), is the subgroup of the mapping class group consisting of elements that act trivially on the homology of the surface. There are three types of elements that naturally arise in studying I(S_g): bounding pair maps, separating twists, and simply intersecting pair maps (SIP-maps). Historically the first two types of elements have been the focus of the literature on I(S_g), while SIP-maps have received relatively little attention until recently, due to an infinite presentation of I(S_g) introduced by Andrew Putman that uses all three types of elements. We will give a topol…

Quasi-isometry type of the metric space derived from the kernel of the Calabi homomorphism

We prove that the set of symmetrized conjugacy classes of the kernel of the Calabi homomorphism on the group of area-preserving diffeomorphisms of the $2$-disk is not quasi-isometric to the half line.

Homomorphisms between mapping class groups

Suppose that $X$ and $Y$ are surfaces of finite topological type, where $X$ has genus $g\geq 6$ and $Y$ has genus at most $2g-1$; in addition, suppose that $Y$ is not closed if it has genus $2g-1$. Our main result asserts that every non-trivial homomorphism $\Map(X) \to \Map(Y)$ is induced by an {\em embedding}, i.e. a combination of forgetting punctures, deleting boundary components and subsurface embeddings. In particular, if $X$ has no boundary then every non-trivial endomorphism $\Map(X)\to\Map(X)$ is in fact an isomorphism. As an application of our main theorem we obtain that, under the s…

Extensions of Johnson's and Morita's homomorphisms that map to finitely generated abelian groups

We extend each higher Johnson homomorphism to a crossed homomorphism from the automorphism group of a finite-rank free group to a finite-rank abelian group. We also extend each Morita homomorphism to a crossed homomorphism from the mapping class group of once-bounded surface to a finite-rank abelian group. This improves on the author's previous results [Algebr. Geom. Topol. 7 (2007):1297-1326]. To prove the first result, we express the higher Johnson homomorphisms as coboundary maps in group cohomology. Our methods involve the topology of nilpotent homogeneous spaces and lattices in nilpotent …

Generalized Johnson homomorphisms for extended N-series

The Johnson filtration of the mapping class group of a compact, oriented surface is the descending series consisting of the kernels of the actions on the nilpotent quotients of the fundamental group of the surface. Each term of the Johnson filtration admits a Johnson homomorphism, whose kernel is the next term in the filtration. In this paper, we consider a general situation where a group acts on a group with a filtration called an "extended N-series". We develop a theory of Johnson homomorphisms in this general setting, including many known variants of the original Johnson homomorphisms as we…

The Johnson homomorphism and its kernel

We give a new proof of a celebrated theorem of Dennis Johnson that asserts that the kernel of the Johnson homomorphism on the Torelli subgroup of the mapping class group is generated by separating twists. In fact, we prove a more general result that also applies to "subsurface Torelli groups". Using this, we extend Johnson's calculation of the rational abelianization of the Torelli group not only to the subsurface Torelli groups, but also to finite-index subgroups of the Torelli group that contain the kernel of the Johnson homomorphism.

On Residualizing Homomorphisms Preserving Quasiconvexity

$H$ is called a $G$-subgroup of a hyperbolic group $G$ if for any finite subset $M\subset G$ there exists a homomorphism from $G$ onto a non-elementary hyperbolic group $G_1$ that is surjective on $H$ and injective on $M$. In his paper in 1993 A. Ol'shanskii gave a description of all $G$-subgroups in any given non-elementary hyperbolic group $G$. Here we show that for the same class of $G$-subgroups the finiteness assumption on $M$ (under certain natural conditions) can be replaced by an assumption of quasiconvexity.

Quasi-homomorphism rigidity with noncommutative targets

As a strengthening of Kazhdan's property (T) for locally compact groups, property (TT) was introduced by Burger and Monod. In this paper, we add more rigidity and introduce property (TTT). This property is suited for the study of rigidity phenomena for quasi-homomorphisms with noncommutative targets. Partially upgrading a result of Burger and Monod, we will prove that SL(n,R) with n at least 3 and their lattices have property (TTT). As a corollary, we generalize the well-known fact that every homomorphism from such a lattice into an amenable group or a hyperbolic group has finite image to the …

Double Johnson filtrations for mapping class groups

We first develop a general theory of Johnson filtrations and Johnson homomorphisms for a group $G$ acting on another group $K$ equipped with a filtration indexed by a "good" ordered commutative monoid. Then, specializing it to the case where the monoid is the additive monoid $\mathbb{N}^2$ of pairs on nonnegative integers, we obtain a theory of double Johnson filtrations and homomorphisms. We apply this theory to the mapping class group $\mathcal{M}$ of a surface $Σ_{g,1}$ with one boundary component, equipped with the normal subgroups $\bar{X}$, $\bar{Y}$ of $π_1(Σ_{g,1})$ associated to a sta…

Geometry of infinitely presented small cancellation groups, Rapid Decay and quasi-homomorphisms

We study the geometry of infinitely presented groups satisfying the small cancelation condition C'(1/8), and define a standard decomposition (called the criss-cross decomposition) for the elements of such groups. We use it to prove the Rapid Decay property for groups with the stronger small cancelation property C'(1/10). As a consequence, the Metric Approximation Property holds for the reduced C*-algebra and for the Fourier algebra of such groups. Our method further implies that the kernel of the comparison map between the bounded and the usual group cohomology in degree 2 has a basis of power…

Two mod-p Johnson filtrations

We consider two mod-p central series of the free group given by Stallings and Zassenhaus. Applying these series to definitions of Dennis Johnson's filtration of the mapping class group we obtain two mod-p Johnson filtrations. Further, we adapt the definition of the Johnson homomorphisms to obtain mod-p Johnson homomorphisms. We calculate the image of the first of these homomorphisms. We give generators for the kernels of these homomorphisms as well. We restrict the range of our mod-p Johnson homomorphisms using work of Morita. We finally prove the announced result of Perron that a rational hom…

First cohomology of pure mapping class groups of big genus one and zero surfaces

We prove that the first integral cohomology of pure mapping class groups of infinite type genus one surfaces is trivial. For genus zero surfaces we prove that not every homomorphism to $\mathbb{Z}$ factors through a sphere with finitely many punctures. In fact we get an uncountable family of such maps.

具有非阿贝尔目标的群的拟仿射与拟二次映射

Quasi-affine and quasi-quadratic maps of groups with non-abelian targets

本文证明了中间拟同态是拟同态的有界扰动，定义并探讨了拟多项式映射的可构造性，同时证明了到无挠双曲群的拟二次映射的有界扰动刚性。

Embedding mapping-class groups of orientable surfaces with one boundary component

Let $S_{g,1,p}$ be an orientable surface of genus $g$ with one boundary component and $p$ punctures. Let $\mathcal{M}_{g,1,p}$ be the mapping-class group of $S_{g,1,p}$ relative to the boundary. We construct homomorphisms $\mathcal{M}_{g,1,p} \to \mathcal{M}_{g',1,(b-1)}$, where $g' \geq 0$ and $b\geq 1$. We proof that the constructed homomorphisms $\M_{g,1,p} \to \M_{g',1,(b-1)}$ are injective. One of these embeddings for $g = 0$ is classic.

群的拟收缩核

Quasi-retracts of groups

本文研究群到其子群的拟收缩核这一特殊拟同态类，给出其代数几何性质、提出拟分裂短正合列理论并建立与拟作用的联系，还得到双曲空间作用继承等几何应用。

The Action homomorphism, quasimorphisms and moment maps on the space of compatible almost complex structures

We extend the definition of Weinstein's Action homomorphism to Hamiltonian actions with equivariant moment maps of (possibly infinite-dimensional) Lie groups on symplectic manifolds, and show that under conditions including a uniform bound on the symplectic areas of geodesic triangles the resulting homomorphism extends to a quasimorphism on the universal cover of the group. We apply these principles to finite dimensional Hermitian Lie groups like Sp(2n,R), reinterpreting the Guichardet-Wigner quasimorphisms, and to the infinite dimensional groups of Hamiltonian diffeomorphisms Ham(M,\om) of cl…

到实代数群的准同态

Quasihomomorphisms to real algebraic groups

本文研究了实线性代数群中的准同态，证明了与离散群中类似但针对实代数群的刚性定理。

On quasihomomorphisms with noncommutative targets

We describe structure of quasihomomorphisms from arbitrary groups to discrete groups. We show that all quasihomomorphisms are 'constructible', i.e., are obtained via certain natural operations from homomorphisms to some groups and quasihomomorphisms to abelian groups. We illustrate this theorem by describing quasihomomorphisms to certain classes of groups. For instance, every unbounded quasihomomorphism to a torsion-free hyperbolic group H is either a homomorphism to a subgroup of H or is a quasihomomorphism to an infinite cyclic subgroup of H.

由Shelukhin拟同态构造的通量同态与双线性形式

Flux homomorphism and bilinear form constructed from Shelukhin's quasimorphism

本文从Shelukhin拟同态出发，在闭连通辛流形的实一阶上同调上构造了辛映射类群不变的交替双线性形式，并由此给出可交换辛同胚通量的约束，推广了Rousseau关于开辛流形的工作。

关于单代数群中格的拟同态刚性

On quasi-homomorphism rigidity for lattices in simple algebraic groups

本文改进了Ozawa的结果，证明了局部域上秩$\geq 2$的单代数群及第二可数局部紧群中的格具有property $(TTT)$，并探讨了具有无限中心的Lie群对该性质的缺失。

Relative homological representations of framed mapping class groups

Let $Σ$ be a surface with either boundary or marked points, equipped with an arbitrary framing. In this note we determine the action of the associated "framed mapping class group" on the homology of $Σ$ relative to its boundary (respectively marked points), describing the image as the kernel of a certain crossed homomorphism related to classical spin structures. Applying recent work of the authors, we use this to describe the monodromy action of the orbifold fundamental group of a stratum of abelian differentials on the relative periods.

全对称集

Totally Symmetric Sets

本文综述全对称集理论及其在对称群、辫群、线性群和映射类群同态问题中的应用。

The Relative Weak Asymptotic Homomorphism Property for Inclusions of Finite von Neumann Algebras

A triple of finite von Neumann algebras $B\subseteq N\subseteq M$ is said to have the relative weak asymptotic homomorphism property if there exists a net of unitary operators $\{u_λ\}_{λ\in Λ}$ in $B$ such that $$\lim_λ|\mathbb{E}}_B(xu_λy)-{\mathbb{E}}_B({\mathbb{E}}_N(x)u_λ{\mathbb{E}}_N(y))\|_2=0$$ for all $x,y\in M$. We prove that a triple of finite von Neumann algebras $B\subseteq N\subseteq M$ has the relative weak asymptotic homomorphism property if and only if $N$ contains the set of all $x\in M$ such that $Bx\subseteq \sum_{i=1}^n x_iB$ for a finite number of elements $x_1,...,x_n$ i…

Commability of groups quasi-isometric to trees

Commability is the finest equivalence relation between locally compact groups such that $G$ and $H$ are equivalent whenever there is a continuous proper homomorphism $G \to H$ with cocompact image. Answering a question of Cornulier, we show that all non-elementary locally compact groups acting geometrically on locally finite simplicial trees are commable, thereby strengthening previous forms of quasi-isometric rigidity for trees. We further show that 6 homomorphisms always suffice, and provide the first example of a pair of locally compact groups which are commable but without commation consis…

Homology and dynamics in quasi-isometric rigidity of once-punctured mapping class groups

In these lecture notes, we combine recent homological methods of Kevin Whyte with older dynamical methods developed by Benson Farb and myself, to obtain a new quasi-isometric rigidity theorem for the mapping class group MCG(S) of a once punctured surface S of genus at least 2: if K is a finitely generated group quasi-isometric to MCG(S) then there is a homomorphism K -> MCG(S) with finite kernel and finite index image. This theorem is joint with Kevin Whyte.

Homomorphisms and rigid isomorphisms of twisted group doubles

We prove several results concerning quasi-bialgebra morphisms $\mathcal{D}^ω(G)\to\mathcal{D}^η(H)$ of twisted group doubles. We take a particular focus on the isomorphisms which are simultaneously isomorphisms $\mathcal{D}(G)\to\mathcal{D}(H)$. All such isomorphisms are shown to be morphisms of quasi-Hopf algebras, and a classification of all such isomorphisms is determined. Whenever $ω\in Z^3(G/Z(G),U(1))$ this suffices to completely describe $\operatorname{Aut}(\mathcal{D}^ω(G))$, the group of quasi-Hopf algebra isomorphisms of $\mathcal{D}^ω(G)$, and so generalizes existing descriptions fo…

Quasi-morphisms on the group of area-preserving diffeomorphisms of the 2-disk via braid groups

Recently Gambaudo and Ghys proved that there exist infinitely many quasi-morphisms on the group ${\rm Diff}_Ω^\infty (D^2, \partial D^2)$ of area-preserving diffeomorphisms of the 2-disk $D^2$. For the proof, they constructed a homomorphism from the space of quasi-morphisms on the braid group to the space of quasi-morphisms on ${\rm Diff}_Ω^\infty (D^2, \partial D^2)$. In this paper, we study the homomorphism and prove its injectivity.

不同秩的自由群是否双不变拟等距？

Are free groups of different ranks bi-invariantly quasi-isometric?

本文证明了装备双不变词度量的有限秩自由群间的同态为拟等距当且仅当其为同构。

Superrigidity from Chevalley groups into acylindrically hyperbolic groups via quasi-cocycles

We prove that every homomorphism from the elementary Chevalley group over a finitely generated unital commutative ring associated with reduced irreducible classical root system of rank at least 2, and ME analogues of such groups, into acylindrically hyperbolic groups has an absolutely elliptic image. This result provides a non-arithmetic generalization of homomorphism superrigidity of Farb--Kaimanovich--Masur and Bridson--Wade.

Coarse median structures and homomorphisms from Kazhdan groups

We study Bowditch's notion of a coarse median on a metric space and formally introduce the concept of a coarse median structure as an equivalence class of coarse medians up to closeness. We show that a group which possesses a uniformly left-invariant coarse median structure admits only finitely many conjugacy classes of homomorphisms from a given group with Kazhdan's property (T). This is a common generalization of a theorem due to Paulin about the outer automorphism group of a hyperbolic group with property (T) as well as of a result of Behrstock-Drutu-Sapir on the mapping class groups of ori…

On abstract homomorphisms of some special unitary groups

We analyze the abstract representations of the groups of rational points of even-dimensional quasi-split special unitary groups associated with quadratic field extensions. We show that, under certain assumptions, such representations have a standard description, as predicted by a conjecture of Borel and Tits. Our method extends the approach introduced by the first author to study abstract representations of Chevalley groups and is based on the construction and analysis of a certain algebraic ring associated to a given abstract representation.

New series in the Johnson cokernels of the mapping class groups of surfaces

In the present paper, we study the Sp-module structure of the cokernel of the Johnson homomorphism of the mapping class groups of surfaces. We detect the Sp-irreducible components with highest weight [1^k] (and [k]) in the cokernel. We also show that the multiplicities of them is equal to one.

Cofinitely Hopfian groups, open mappings and knot complements

A group $Γ$ is defined to be cofinitely Hopfian if every homomorphism $Γ\toΓ$ whose image is of finite index is an automorphism. Geometrically significant groups enjoying this property include certain relatively hyperbolic groups and many lattices. A knot group is cofinitely Hopfian if and only if the knot is not a torus knot. A free-by-cyclic group is cofinitely Hopfian if and only if it has trivial centre. Applications to the theory of open mappings between manifolds are presented.

Constructive homomorphisms for classical groups

Let Omega be a quasisimple classical group in its natural representation over a finite vector space V, and let Delta be its normaliser in the general linear group. We construct the projection from Delta to Delta/Omega and provide fast, polynomial-time algorithms for computing the image of an element. Given a discrete logarithm oracle, we also represent Delta/Omega as a group with at most 3 generators and 6 relations. We then compute canonical representatives for the cosets of Omega. A key ingredient of our algorithms is a new, asymptotically fast method for constructing isometries between spac…

On localizations of quasi-simple groups with given countable center

A group homomorphism $i: H \to G$ is a localization of $H$ if for every homomorphism $φ: H\rightarrow G$ there exists a unique endomorphism $ψ: G\rightarrow G$, such that $i ψ=φ$ (maps are acting on the right). Göbel and Trlifaj asked in \cite[Problem 30.4(4), p. 831]{GT12} which abelian groups are centers of localizations of simple groups. Approaching this question we show that every countable abelian group is indeed the center of some localization of a quasi-simple group, i.e. a central extension of a simple group. The proof uses Obraztsov and Ol'shanskii's construction of infinite simple gr…

A finiteness property on monodromies of holomorphic families

Given a Kahler group, we study the set of homomorphisms from this group to the mapping class group which can be realized as the monodromy of a holomorphic family of curves.

Approximate Representations and Approximate Homomorphisms

Approximate algebraic structures play a defining role in arithmetic combinatorics and have found remarkable applications to basic questions in number theory and pseudorandomness. Here we study approximate representations of finite groups: functions f:G -> U_d such that Pr[f(xy) = f(x) f(y)] is large, or more generally Exp_{x,y} ||f(xy) - f(x)f(y)||^2$ is small, where x and y are uniformly random elements of the group G and U_d denotes the unitary group of degree d. We bound these quantities in terms of the ratio d / d_min where d_min is the dimension of the smallest nontrivial representation o…