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

Abstract

Recently Gambaudo and Ghys proved that there exist infinitely many quasi-morphisms on the group ${\mathrm{Diff}}_\Omega ^\infty (D^2, \partial D^2)$ 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 ${\mathrm{Diff}}_\Omega ^\infty (D^2, \partial D^2)$. In this paper, we study this homomorphism and prove its injectivity.

1. Introduction

For a group $G$, a function $\phi \colon G\to {\mathbb{R}}$ is called a quasi-morphism if the real valued function on $G\times G$ defined by

$$\begin{equation*} (g, h)\mapsto \phi (gh)-\phi (g)-\phi (h) \end{equation*}$$

is bounded. The real number

$$\begin{equation*} D(\phi ) =\sup _{g, h\in G} |\phi (gh)-\phi (g)-\phi (h)| \end{equation*}$$

is called the defect of $\phi$. We denote the ${\mathbb{R}}$-vector space of quasi-morphisms on the group $G$ by $\hat{Q} (G)$. By definition, bounded functions on groups are quasi-morphisms. Hence we denote the set of bounded functions on the group $G$ by $C_b^1(G; {\mathbb{R}})$ and consider the quotient space $Q(G)=\hat{Q}(G)/C_b^1(G; {\mathbb{R}})$. A quasi-morphism $\phi \colon G\to {\mathbb{R}}$ is said to be homogeneous if the equation

$$\begin{equation*} \phi (g^p)=p\ \phi (g) \end{equation*}$$

holds for any $g\in G$ and $p\in {\mathbb{Z}}$. For any quasi-morphism $\phi$, a homogeneous quasi-morphism $\tilde{\phi }$ is defined by setting

$$\begin{equation*} \tilde{\phi }(g)=\lim _{p\to \infty }\frac{1}{p}\phi (g^p). \end{equation*}$$

The limit always exists for each element $g$ of $G$. The new function $\tilde{\phi }$ is in fact a quasi-morphism equal to the original quasi-morphism $\phi$ as an element of $Q(G)$. Thus we can identify the vector space of homogeneous quasi-morphisms on the group $G$ with $Q(G)$. Homogeneous quasi-morphisms are invariant under conjugation. Therefore we are interested in $Q(G)$ rather than $\hat{Q}(G)$.

Let ${\mathrm{Diff}}_\Omega ^\infty (D^2, \partial D^2)$ be the group of area-preserving $C^\infty$-diffeomorphisms of the $2$-disk $D^2$, which are the identity on a neighborhood of the boundary. On the vector space $Q({\mathrm{Diff}}_\Omega ^\infty (D^2, \partial D^2))$, the following theorem is known.

Theorem 1.1 (Entov-Polterovich 3, Gambaudo-Ghys 5).

The vector space $Q({\mathrm{Diff}}_\Omega ^\infty (D^2, \partial D^2))$ is infinite dimensional.

To prove Theorem 1.1, Entov and Polterovich explicitly constructed uncountably many quasi-morphisms on ${\mathrm{Diff}}_\Omega ^\infty (D^2, \partial D^2)$, which are linearly independent. After that Gambaudo and Ghys constructed countably many quasi-morphisms on ${\mathrm{Diff}}_\Omega ^\infty (D^2, \partial D^2)$ using a different idea, which is to consider the suspension of area-preserving diffeomorphisms of the disk and average the value of the signature of the braids appearing in the suspension. By generalizing their strategy Brandenbursky 1 defined the homomorphism

$$\begin{equation*} \Gamma _n\colon Q(P_n(D^2))\to Q({\mathrm{Diff}}_\Omega ^\infty (D^2, \partial D^2)), \end{equation*}$$

which we review in Section 2. Here, $P_n(D^2)$ denotes the pure braid group on $n$-strands.

Let $B_n(D^2)$ be the braid group on $n$-strands. The natural inclusion $i\colon P_n(D^2)\to B_n(D^2)$ induces the homomorphism $Q(i)\colon Q(B_n(D^2))\to Q(P_n(D^2))$. In this paper, we study the homomorphism $\Gamma _n$ and prove the following theorem.

Theorem 1.2.

The composition

$$\begin{equation*} \Gamma _n\circ Q(i)\colon Q(B_n(D^2))\to Q({\mathrm{Diff}}_\Omega ^\infty (D^2, \partial D^2)) \end{equation*}$$

is injective.

2. Gambaudo and Ghys’ construction and proof of the main theorem

In this section, we review Gambaudo and Ghys’ construction 5 of quasi-morphisms on the group ${\mathrm{Diff}}_\Omega ^\infty (D^2, \partial D^2)$ in a generalized form and prove Theorem 1.2.

Let $X_n(D^2)$ be the configuration space of ordered $n$-tuples in the $2$-disk $D^2$ and $x^0=(x _1^0, \dots , x_n^0)$ its base point. For any $g\in {\mathrm{Diff}}_\Omega ^\infty (D^2, \partial D^2)$ and for almost all $x=(x _1, \dots , x_n)\in X_n(D^2)$, we define the pure braid $\gamma (g; x)$ as the following. First we set the loop $l(g; x)\colon [0, 1]\to X_n(D^2)$ by

$$\begin{equation*} l(g; x)(t)=\left\{ \begin{array}{ll} \{ (1-3t)x_i^0+3tx_i\} &\displaystyle (0\leq t\leq \frac{1}{3}),\\[6.00006pt] \{ g_{3t-1}(x_i)\} &\displaystyle (\frac{1}{3}\leq t\leq \frac{2}{3}),\\[6.00006pt] \{ (3-3t)g(x_i)+(3t-2)x_i^0\} &\displaystyle (\frac{2}{3}\leq t\leq 1), \\\end{array}\right. \end{equation*}$$

where $\{ g_t\}_{t\in [0, 1]}$ is a Hamiltonian isotopy such that $g_0$ is the identity and $g_1=g$. We define the pure braid $\gamma (g; x)$ to be the braid represented by the loop $l(g; x)$. For almost every $x$, the braid $\gamma (g; x)$ is well-defined. Furthermore, the braid $\gamma (g; x)$ is independent of the choice of the flow $\{ g_t\}$. This is because of the fact that the group ${\mathrm{Diff}}_\Omega ^\infty (D^2, \partial D^2)$ is contractible, which is easily proved from the contractibility of the diffeomorphism group ${\mathrm{Diff}}^\infty (D^2, \partial D^2)$ of $D^2$ 8 and the homotopy equivalence between ${\mathrm{Diff}}^\infty (D^2, \partial D^2)$ and ${\mathrm{Diff}}_\Omega ^\infty (D^2, \partial D^2)$ 7. For a quasi-morphism $\phi$ on the pure braid group $P_n(D^2)$ on $n$-strands, we define the function $\hat{\Gamma }_n(\phi )\colon {\mathrm{Diff}}_\Omega ^\infty (D^2, \partial D^2)\to {\mathbb{R}}$ by

$$\begin{equation*} \hat{\Gamma }_n(\phi )(g)=\int _{x\in X_n(D^2)}\phi (\gamma (g; x))dx. \end{equation*}$$

For any $\phi \in Q(P_n(D^2))$ and $g\in {\mathrm{Diff}}_\Omega ^\infty (D^2, \partial D^2)$ the function $\phi (\gamma (g; \cdot ))$ is integrable and thus the map $\hat{\Gamma }_n\colon \hat{Q}(P_n(D^2))\to \hat{Q}({\mathrm{Diff}}_\Omega ^\infty (D^2, \partial D^2))$ is well-defined 2. The obtained function $\hat{\Gamma }_n(\phi )\colon {\mathrm{Diff}}_\Omega ^\infty (D^2, \partial D^2)\to {\mathbb{R}}$ is also a quasi-morphism, and the map $\hat{\Gamma }_n\colon \hat{Q}(P_n(D^2))\to \hat{Q}({\mathrm{Diff}}_\Omega ^\infty (D^2, \partial D^2))$ is clearly ${\mathbb{R}}$-linear. Moreover, it is easily checked that any bounded function on $P_n(D^2)$ is mapped to a bounded function on ${\mathrm{Diff}}_\Omega ^\infty (D^2, \partial D^2)$, and thus the homomorphism

$$\begin{equation*} \hat{\Gamma }_n\colon \hat{Q}(P_n(D^2)) \to \hat{Q} ({\mathrm{Diff}}_\Omega ^\infty (D^2,\partial D^2)) \end{equation*}$$

induces the homomorphism $\Gamma _n \colon Q(P_n(D^2)) \to Q({\mathrm{Diff}}_\Omega ^\infty (D^2, \partial D^2))$.

Remark 2.1.

We see that the homomorphism $\Gamma _n \colon Q(P_n(D^2))\! \to \! Q({\mathrm{Diff}}_\Omega ^\infty (D^2, \partial D^2))$ maps the classical linking number homomorphism ${\mathrm{lk}}_n\colon B_n(D^2)\to {\mathbb{R}}$ on the braid group to a homomorphism on ${\mathrm{Diff}}_\Omega ^\infty (D^2, \partial D^2)$. In fact, the image of ${\mathrm{lk}}\colon B_n(D^2)\to {\mathbb{R}}$ by the homomorphism $\Gamma _n({\mathrm{lk}}_n)$ coincides with a constant multiple of the classical Calabi homomorphism on ${\mathrm{Diff}}_\Omega ^\infty (D^2, \partial D^2)$ 4, and in this sense quasi-morphisms obtained in this way can be considered as generalizations of the Calabi homomorphism. By an argument of Brandenbursky which verifies that the homomorphism $\Gamma \colon \hat{Q}(P_n)\to \hat{Q}({\mathrm{Diff}}_\Omega ^\infty (D^2, \partial D^2))$ is well-defined, it is observed that quasi-morphisms obtained by the homomorphism $\hat{\Gamma }_n \colon \hat{Q}(P_n(D^2)) \to \hat{Q}({\mathrm{Diff}}_\Omega ^\infty (D^2, \partial D^2))$ can be defined on the group of area-preserving $C^1$-diffeomorphisms of $D^2$, as well as the Calabi homomorphism.

Now we are ready to prove Theorem 1.2.

Proof of Theorem 1.2.

Let us suppose that a homogeneous quasi-morphism $\phi \in \hat{Q}(B_n(D^2))$ is non-trivial. Then there exists a braid $\beta \in B_n(D^2)$ such that $\phi (\beta )\neq 0$. We may assume that $\beta$ is pure. It is sufficient to prove that the homogeneous quasi-morphism $\hat{\Gamma }_n(\phi )\in \hat{Q}({\mathrm{Diff}}_\Omega ^\infty (D^2, \partial D^2))$ is also non-trivial. That is, there exists an area-preserving diffeomorphism $g\in {\mathrm{Diff}}_\Omega ^\infty (D^2, \partial D^2)$ such that

$$\begin{equation*} \lim _{p\to \infty }\frac{1}{p}\Gamma _n(\phi )(g^p)\neq 0. \end{equation*}$$

Let $A_{i, j}$ be the pure braid which twists only the $i$-th and the $j$-th strands for $1\leq i<j\leq n$ (see Figure 1).

Since the braid $\beta$ is pure, it can be written as a composition of $A_{i, j}$’s and their inverses. We take $n$ disjoint subsets $U_i$ of $D^2$. Furthermore, for a pair of $(i, j)$, we take subsets $V_{i, j}$ and $W_{i, j}$ of $D^2$ such that $U_i\cup U_j\subset W_{i, j}\subset V_{i, j}$, $U_k\cap V_{i, j}=\emptyset$ if $k\neq i, j$ and $V_{i, j}, W_{i, j}$ are diffeomorphic to $D^2$. Let $\{ h_t\}_{t\in [0, 1]}$ be a path in ${\mathrm{Diff}}_\Omega ^\infty (D^2, \partial D^2)$ such that the support of $h_t$ is contained in the interior of $V_{i, j}$ and rotates $W_{i, j}$ once. Taking paths $\{ h_t\}$ constructed above for all the $A_{i, j}$’s which present $\beta$ and composing them, we have a path $\{ g_t\}_{t\in [0, 1]}$ in ${\mathrm{Diff}}_\Omega ^\infty (D^2, \partial D^2)$ with $g_0=id$ which twists the $U_i$’s in the form of the pure braid $\beta$. If we set $g=g_1$, then $g$ is the identity on the $U_i$’s and $\gamma (g; (x_1, \dots , x_n))=\beta$ for $x_i\in U_i$. Then by setting $U=U_1\cup \dots \cup U_n$, we have

$$\begin{align*} &\quad \lim _{p\to \infty }\frac{1}{p}\hat{\Gamma } _n(\phi )(g^p) \\ &=\lim _{p\to \infty }\frac{1}{p} \left(\int _{x\in X_n(U)}\phi (\gamma (g^p; x))dx +\int _{x\in X_n(D^2)\setminus X_n(U)}\phi (\gamma (g^p ; x))dx\right) \\ &=\int _{x\in X_n(U)}\phi (\gamma (g; x))dx +\lim _{p\to \infty }\frac{1}{p}\int _{x\in X_n(D^2)\setminus X_n(U)}\phi (\gamma (g^p ; x))dx . \end{align*}$$

If we denote the first term of the equation by $Y$ and set $a_i={\mathrm{area}}(U_i)$ and $[n]=\{ 1, \dots , n\}$, then $Y$ is written as

$$\begin{equation*} \int _{x\in X_n(U)}\phi (\gamma (g; x))dx=\sum _{F\colon [n]\to [n]} \left( \prod _{i=1}^na_{F(i)}\right) x_F, \end{equation*}$$

where $x_F=\phi (\gamma (g; x))$ for $x$ in the case when each $x_i$ is in $U_{F(i)}$. The real numbers $x_F$ have the following properties:

(i)

For two maps $F\text{ and }G\colon [n]\to [n]$, if $\# F^{-1}(i)=\# G^{-1}(i)$ for each $1\leq i\leq n$, then $x_F=x_G$.

(ii)

If a map $F\colon [n]\to [n]$ is bijective, then $x_F$ is non-zero.

Property (i) follows from the invariance of $\phi$ under conjugation, and property (ii) follows because $\phi (\beta )$ is non-zero. Therefore, the coefficient of $a_1\dots a_n$ in $Y$ is non-zero. Since the polynomial $Y$ is not identically $0$, we can choose the $a_i$’s such that $Y$ is non-zero.

Note that if we replace the $a_i$’s by bigger ones, by fixing the ratio of any two of them the term $Y$ stays non-zero. On the other hand, the value $\phi (\gamma (g ; x))$ is bounded because of the construction of $g$, and we thus have

$$\begin{equation*} \lim _{p\to \infty }\frac{1}{p}\int _{x\notin X_n(U)}\phi (\gamma (g^p ; x))dx \to 0 \quad \text{(as}\quad a_1+\dots +a_n\to {\mathrm{area}}(D^2)\text{)}. \end{equation*}$$

This completes the proof.

■

As we noted in Remark 2.1, the homomorphism $\hat{\Gamma }_n$ maps any homomorphism on $P_n(D^2)$ to a homomorphism on ${\mathrm{Diff}}_\Omega ^\infty (D^2, \partial D^2)$. Hence the homomorphism

$$\begin{equation*} Q(P_n(D^2))/H^1(P_n(D^2); {\mathbb{R}})\to Q({\mathrm{Diff}}_\Omega ^\infty (D^2, \partial D^2))/H^1({\mathrm{Diff}}_\Omega ^\infty (D^2, \partial D^2); {\mathbb{R}}) \end{equation*}$$

is also induced. By an argument similar to the proof of Theorem 1.2, the following proposition holds.

Proposition 2.2.

The map

$$\begin{equation*} Q(B_n(D^2))/H^1(B_n(D^2); {\mathbb{R}})\to Q({\mathrm{Diff}}_\Omega ^\infty (D^2, \partial D^2))/H^1({\mathrm{Diff}}_\Omega ^\infty (D^2, \partial D^2); {\mathbb{R}}) \end{equation*}$$

induced by the composition $\Gamma _n\circ Q(i)\colon Q(B_n(D^2))\to Q({\mathrm{Diff}}_\Omega ^\infty (D^2, \partial D^2))$ is injective.

The homomorphism $\Gamma _n \colon Q(P_n(D^2))\to Q({\mathrm{Diff}}_\Omega ^\infty (D^2, \partial D^2))$ can also be defined for the $2$-sphere $S^2$ instead of $D^2$ as Gambaudo and Ghys mentioned in their paper. Let ${\mathrm{Diff}}_\Omega ^\infty (S^2)_0$ be the identity component of the group of area-preserving diffeomorphisms of $S^2$. Then we can choose a pure braid $\gamma (g; x)\in P_n(S^2)$ for any $g\in {\mathrm{Diff}}_\Omega ^\infty (S^2)_0$ and for almost every $x\in X_n(S^2)$ as in the case of the $2$-disk. Since the group ${\mathrm{Diff}}_\Omega ^\infty (S^2)_0$ is homotopy equivalent to $SO(3)$ 7, 8 and its fundamental group has order $2$, for any element $g$ of ${\mathrm{Diff}}_\Omega ^\infty (S^2)_0$ there exist two homotopy classes of paths connecting the identity and $g$ in ${\mathrm{Diff}}_\Omega ^\infty (S^2)_0$. However, for any homogeneous quasi-morphism $\phi$ on $P_n(S^2)$, the value $\phi (\gamma (g; x))$ is independent of the choice of the path. In fact, the braid obtained from a path which represents the generator of $\pi _1({\mathrm{Diff}}_\Omega ^\infty (S^2)_0)$ has order $2$ and is in the center of $P_n(S^2)$. Hence the homomorphism $\Gamma _n\colon Q(P_n(S^2))\to Q({\mathrm{Diff}}_\Omega ^\infty (S^2)_0)$ is defined. Since the braid group $B_n(S^2)$ of the $2$-sphere on $n$-strands can be considered as a quotient group of the braid group $B_n(D^2)$, by an argument similar to the proof of Theorem 1.2, we obtain the following theorem.

Theorem 2.3.

The composition

$$\begin{equation*} \Gamma _n\circ Q(i)\colon Q(B_n(S^2))\to Q({\mathrm{Diff}}_\Omega ^\infty (S^2)_0) \end{equation*}$$

is injective.

The homomorphism $Q(i)$ in the statement of Theorem 2.3 is the one induced from the inclusion $i\colon P_n(S^2)\to B_n(S^2)$.

3. Kernel of the homomorphism $\Gamma _n$

The homomorphism $\Gamma _n\colon Q(P_n (D^2))\to Q({\mathrm{Diff}}_\Omega ^\infty (D^2, \partial D^2))$ itself is not injective although Theorem 1.2 holds. In this section we study the kernel of the homomorphism $\Gamma _n$.

Let $G$ be a group and $H$ its finite index subgroup. We denote by $\overline{\beta }$ the image of an element $\beta \in G$ by the natural projection $G\to G/H$. For each left coset $\sigma \in G/H$ of $G$ modulo $H$, we fix an element $\gamma _\sigma \in G$ such that $\overline{\gamma _\sigma }=\sigma$ and for any $\phi \in \hat{Q}(H)$ define the function $\hat{{\mathcal{T}}}(\phi )\colon G\to {\mathbb{R}}$ by

$$\begin{equation*} \hat{{\mathcal{T}}}(\phi )(\beta )= \frac{1}{(G:H)}\sum _{\sigma \in G/H} \phi ({\gamma _{\overline{\beta \gamma _{\sigma }}}}^{-1} \beta \gamma _{\sigma }). \end{equation*}$$

Since ${\gamma _{\overline{\beta \gamma _{\sigma }}}}^{-1} \beta \gamma _{\sigma }$ is in $H$, the function $\hat{{\mathcal{T}}}(\phi )$ is well-defined on $G$.

Lemma 3.1.

For any quasi-morphism $\phi$ on $H$, the function $\hat{{\mathcal{T}}}(\phi )\colon G\to {\mathbb{R}}$ is also a quasi-morphism.

Proof.

Since the equality

$$\begin{equation*} {\gamma _{\overline{\beta _1\beta _2\gamma _\sigma }}}^{-1} \beta _1\beta _2 \gamma _\sigma =({\gamma _{\overline{\beta _1\beta _2\gamma _\sigma }}}^{-1} \beta _1\gamma _{\overline{\beta _2\gamma _\sigma }}) ({\gamma _{\overline{\beta _2\gamma _\sigma }}}^{-1}\beta _2\gamma _\sigma ) \end{equation*}$$

holds, we have the inequality

$$\begin{align*} &\quad |\hat{{\mathcal{T}}}(\phi )(\beta _1\beta _2)-\hat{{\mathcal{T}}}(\phi )(\beta _1)-\hat{{\mathcal{T}}}(\phi )(\beta _2)| \\ &=\frac{1}{(G:H)}\Bigg |\sum _{\sigma \in G/H}\Big \{ \phi (({\gamma _{\overline{\beta _1\beta _2\gamma _\sigma }}}^{-1} \beta _1\gamma _{\overline{\beta _2\gamma _\sigma }}) ({\gamma _{\overline{\beta _2\gamma _\sigma }}}^{-1}\beta _2\gamma _\sigma )) \\ &\qquad \qquad \qquad \qquad \qquad -\phi ({\gamma _{\overline{\beta _1\gamma _{\sigma }}}}^{-1} \beta _1\gamma _{\sigma }) -\phi ({\gamma _{\overline{\beta _2\gamma _{\sigma }}}}^{-1} \beta _2\gamma _{\sigma }) \Big \} \Bigg | \\ &=\frac{1}{(G:H)}\Bigg |\sum _{\sigma \in G/H}\Big \{ \phi (({\gamma _{\overline{\beta _1\beta _2\gamma _\sigma }}}^{-1} \beta _1\gamma _{\overline{\beta _2\gamma _\sigma }}) ({\gamma _{\overline{\beta _2\gamma _\sigma }}}^{-1}\beta _2\gamma _\sigma )) \\ &\qquad \qquad \qquad \qquad \qquad -\phi ({\gamma _{\overline{\beta _1\beta _2\gamma _\sigma }}}^{-1} \beta _1\gamma _{\overline{\beta _2\gamma _\sigma }}) -\phi ({\gamma _{\overline{\beta _2\gamma _{\sigma }}}}^{-1} \beta _2\gamma _{\sigma }) \Big \} \Bigg | \\ &\leq D(\phi ) . \end{align*}$$

Hence the function $\hat{{\mathcal{T}}}(\phi )\colon G\to {\mathbb{R}}$ is also a quasi-morphism.

■

The map $\hat{{\mathcal{T}}}\colon \hat{Q}(H)\to \hat{Q}(G)$ is clearly ${\mathbb{R}}$-linear and induces a homomorphism ${\mathcal{T}}\colon Q(P_n(D^2)) \to Q(B_n(D^2))$. Furthermore, the following proposition holds.

Proposition 3.2.

The homomorphism ${\mathcal{T}}\colon Q(H) \to Q(G)$ is independent of the choice of $\gamma _\sigma$’s.

Proof.

Suppose that $\phi$ is a homogeneous quasi-morphism on $H$. If an element $\beta$ is in $H$, then $\overline{\gamma _\sigma \beta }=\sigma$ for each $\sigma \in G/H$. For any $\beta \in G$ there exists an integer $k$ such that $\beta ^k$ is in $H$ and we have

$$\begin{align} \lim _{p\to \infty } \frac{1}{p}\hat{{\mathcal{T}}}(\phi )(\beta ^p) &=\lim _{p'\to \infty } \frac{1}{kp'}\hat{{\mathcal{T}}}(\phi )(\beta ^{kp'}) \\ &=\lim _{p'\to \infty } \frac{1}{(G:H)kp'}\sum _{\sigma \in G/H} \phi (\gamma _\sigma ^{-1}\beta ^k\gamma _\sigma )^{p'} \\ &=\frac{1}{(G:H)k}\sum _{\sigma \in G/H}\phi (\gamma _\sigma ^{-1}\beta ^k\gamma _\sigma ). {\tag{3.1}} \end{align}$$

Since $\phi$ is invariant under conjugations in $H$, the value $\phi (\gamma _\sigma ^{-1}\beta ^k\gamma _\sigma )$ depends only on $\sigma$.

■

Let $Q(i)\colon Q(G)\to Q(H)$ be the homomorphism induced by the inclusion $i\colon H\to G$. As a corollary to equality (3.1), we have the following.

Corollary 3.3.

The composition ${\mathcal{T}}\circ Q(i)\colon Q(G)\to Q(G)$ is the identity on $Q(G)$. Furthermore, we have the decomposition

$$\begin{equation*} Q(H)={\mathrm{Ker}}({\mathcal{T}})\oplus {\mathrm{Im}}(Q(i)) \end{equation*}$$

as vector spaces.

Remark 3.4.

Of course, the homomorphism $\hat{{\mathcal{T}}}(\phi )\colon G\to {\mathbb{R}}$ can be defined using the right coset $H\backslash G$ instead of $G/H$ by

$$\begin{equation*} \hat{{\mathcal{T}}}(\phi )(\beta )= \frac{1}{(G:H)}\sum _{\sigma \in G/H} \phi (\gamma _\sigma \beta {\gamma _{\overline{\gamma _\sigma \beta }}}^{-1}). \end{equation*}$$

By an argument similar to the proof of Lemma 3.1 and Proposition 3.2, it is verified that this alternative definition is also well-defined and induces the same homomorphism ${\mathcal{T}}\colon Q(H) \to Q(G)$.

Remark 3.5.

The homomorphism ${\mathcal{T}}\colon Q(H)\to Q(G)$ is just a straightforward generalization of the transfer map, and it is also introduced in 6 and 9.

Since the pure braid groups $P_n(D^2)$ and $P_n(S^2)$ are finite index subgroups of the braid groups $B_n(D^2)$ and $B_n(S^2)$, respectively, the homomorphisms

$$\begin{equation*} {\mathcal{T}}\colon Q(P_n(D^2))\to Q(B_n(D^2)) \quad \text{ and }\quad {\mathcal{T}}\colon Q(P_n(S^2))\to Q(B_n(S^2)) \end{equation*}$$

can be defined and Corollary 3.3 is true for $G=B_n(D^2), H=P_n(D^2)$ and $G=B_n(S^2), H=P_n(S^2)$, respectively.

The following proposition is the main result of this section.

Proposition 3.6.

The composition

$$\begin{equation*} \Gamma _n\circ Q(i)\circ {\mathcal{T}}\colon Q(P_n(D^2))\to Q({\mathrm{Diff}}_\Omega ^\infty (D^2, \partial D^2)) \end{equation*}$$

coincides with $\Gamma _n$. In particular, ${\mathrm{Ker}}(\Gamma _n)={\mathrm{Ker}}({\mathcal{T}})$ and ${\mathrm{Im}}(\Gamma _n)={\mathrm{Im}}(\Gamma _n\circ Q(i))$.

Proof.

Let ${\mathfrak{S}}_n$ be the symmetric group of $n$ symbols. By equality (3.1), for any homogeneous quasi-morphism $\phi \in Q(P_n(D^2))$ and any area-preserving diffeomorphism $g\in {\mathrm{Diff}}_\Omega ^\infty (D^2, \partial D^2)$,

$$\begin{align} \lim _{p\to \infty }\frac{1}{p} \hat{\Gamma }_n\circ Q(i)\circ \hat{{\mathcal{T}}} (\phi )(g^p) =\frac{1}{n!}\sum _{\sigma \in {\mathfrak{S}}_n} \lim _{p\to \infty }\frac{1}{p} \int _{x\in X_n(D^2)} \phi (\gamma _\sigma \gamma (g^p; x)\gamma ^{-1}_\sigma )dx. {\tag{3.2}} \end{align}$$

For any $\sigma \in {\mathfrak{S}}_n$ and almost all $x\in D^2$, we set the path $l\colon [0, 1]\to X_n(D^2)$ by

$$\begin{equation*} l(t)=\left\{ \begin{array}{ll} \{ (1-2t)x_i^0+2tx_i\} &\displaystyle (0\leq t\leq \frac{1}{2}), \\[6.00006pt] \{ (2-2t)x_i+(2t-1)x_{\sigma (i)}^0\} &\displaystyle (\frac{1}{2}\leq t\leq 1). \\\end{array}\right. \end{equation*}$$

Considering the path $l$ as a loop in the quotient space $X_n(D^2)/{\mathfrak{S}}_n$, we define the braid $\beta (\sigma ; x)$ to be the braid represented by the loop $l$. Then by definition,

$$\begin{equation*} \beta (\sigma ; x)\gamma (g; \sigma ^{-1}(x)) \beta (\sigma ; g_\ast x)^{-1}=\gamma (g; x), \end{equation*}$$

where the symmetric group ${\mathfrak{S}}_n$ acts on $X_n(D^2)$ by the permutation

$$\begin{equation*} \sigma (x_1, \dots , x_n)=(x_{\sigma (1)}, \dots , x_{\sigma (n)}). \end{equation*}$$

Since the homomorphism ${\mathcal{T}}\colon Q(P_n(D^2))\to Q(B_n(D^2))$ is defined independently of the choice of braids $\gamma _\sigma$’s, we may choose $\gamma _\sigma$ to be $\beta (\sigma ; x)$. Hence we have

$$\begin{align*} \gamma _\sigma \gamma (g; \sigma ^{-1}(x))\gamma _\sigma ^{-1} &=\beta (\sigma ;x)\gamma (g; \sigma ^{-1}(x))\beta (\sigma ;x)^{-1} \\ &=\gamma (g; x)\beta (\sigma ; g_\ast (x))\beta (\sigma ; x)^{-1}. \end{align*}$$

Since the function $\phi (\beta (\sigma ; \cdot ))\colon D^2\to {\mathbb{R}}$ is bounded on $D^2$, we have

$$\begin{align*} &\quad \lim _{p\to \infty }\frac{1}{p} \int _{x\in X_n(D^2)} \phi (\gamma _\sigma \gamma (g^p; x)\gamma ^{-1}_\sigma )dx \\ &=\lim _{p\to \infty }\frac{1}{p} \int _{x\in X_n(D^2)} \phi (\gamma _\sigma \gamma (g^p; \sigma ^{-1}(x))\gamma ^{-1}_\sigma )dx \\ &=\lim _{p\to \infty }\frac{1}{p} \int _{x\in X_n(D^2)} \phi (\gamma (g^p; x))dx. \end{align*}$$

Therefore, by equality (3.2),

$$\begin{align*} \lim _{p\to \infty }\frac{1}{p} \hat{\Gamma }_n\circ Q(i)\circ \hat{{\mathcal{T}}} (\phi )(g^p) &=\frac{1}{n!}\sum _{\sigma \in {\mathfrak{S}}_n} \lim _{p\to \infty }\frac{1}{p} \int _{x\in X_n(D^2)} \phi (\gamma (g^p; x))dx \\ &=\lim _{p\to \infty }\frac{1}{p}\hat{\Gamma }_n(\phi )(g^p), \end{align*}$$

and thus we have $\Gamma _n\circ Q(i)\circ {\mathcal{T}}=\Gamma _n$.

Then obviously ${\mathrm{Ker}}({\mathcal{T}})\subseteq {\mathrm{Ker}}(\Gamma _n)$ and ${\mathrm{Im}}(\Gamma _n)={\mathrm{Im}}(\Gamma _n\circ Q(i))$ hold. If $\phi \in {\mathrm{Ker}}(\Gamma _n)$, then

$$\begin{equation*} \Gamma _n\circ Q(i)\circ {\mathcal{T}}(\phi )=\Gamma _n(\phi )=0, \end{equation*}$$

and hence ${\mathcal{T}}(\phi )=0$ by Theorem 1.2. Thus we have ${\mathrm{Ker}}(\Gamma _n)\subseteq {\mathrm{Ker}}({\mathcal{T}})$.

■

Remark 3.7.

Proposition 3.6 also holds for $P_n(S^2)$ and ${\mathrm{Diff}}_\Omega ^\infty (S^2)_0$ instead of $P_n(D^2)$ and ${\mathrm{Diff}}_\Omega ^\infty (D^2, \partial D^2)$, respectively.

Figure 1.

Pure braid $A_{i, j}$