Quasi-morphisms on the group of area-preserving diffeomorphisms of the $2$-disk via braid groups
By Tomohiko Ishida
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
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
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.
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 Reference 1 defined the homomorphism
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.
2. Gambaudo and Ghys’ construction and proof of the main theorem
In this section, we review Gambaudo and Ghys’ construction Reference 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
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 $D^2$Reference 8 and the homotopy equivalence between ${\mathrm{Diff}}^\infty (D^2, \partial D^2)$ and ${\mathrm{Diff}}_\Omega ^\infty (D^2, \partial D^2)$Reference 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
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 Reference 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
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
is also induced. By an argument similar to the proof of Theorem 1.2, the following proposition holds.
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 $SO(3)$Reference 7, Reference 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.
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
Since ${\gamma _{\overline{\beta \gamma _{\sigma }}}}^{-1} \beta \gamma _{\sigma }$ is in $H$, the function $\hat{{\mathcal{T}}}(\phi )$ is well-defined on $G$.
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.
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 (Equation 3.1), we have the following.
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
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.
Acknowledgments
This work is the main part of the author’s doctoral thesis at the University of Tokyo, under the supervision of Professor Takashi Tsuboi. The author wishes to thank him for much helpful advice. The author also thanks Professors Étienne Ghys and Shigeyuki Morita for their warm encouragement and Professor Shigenori Matsumoto for valuable suggestions. The author is very grateful to the referee for a careful reading and for pointing out errors in the manuscript. The author was supported by JSPS Research Fellowships for Young Scientists (23$\cdot$1352).
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