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

TL;DR

This paper demonstrates that the metric space formed from the kernel of the Calabi homomorphism on area-preserving diffeomorphisms of the 2-disk has a complex geometric structure, not resembling a simple half-line.

Contribution

It establishes the quasi-isometry type of the metric space derived from the kernel of the Calabi homomorphism, revealing its non-linear geometric nature.

Findings

The space is not quasi-isometric to the half line.

The structure of the kernel's conjugacy classes is complex.

The result provides insight into the geometry of symplectic diffeomorphism groups.

Abstract

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.

Equations

$$d([f],[g])=\log \max \{q_{\{g\}}(f),q_{\{f\}}(g)\}.$$

$$q_{\{f\}}(h)\le q_{\{f\}}(g)q_{\{g\}}(h)$$

$$D^2=\{(x,y)\in \mathbb{R};x^2+y^2\le 1\}\text{ and }\Omega =\frac{1}{\pi }dx\wedge dy$$

$$\mathrm{Cal}:{\mathrm{Diff}}_{\Omega }^{\infty }(D^2,\partial D^2)\to \mathbb{R}$$

$$D(\varphi )=\underset{g,h\in G}{\sup }|\varphi (gh)-\varphi (g)-\varphi (h)|$$

$$\tilde{\varphi }(g)=\underset{p\to \infty }{\lim }\frac{1}{p}\varphi (g^p).$$

$$Q(G,K)=\{\varphi \in Q(G);\varphi \text{ is bounded on }K\}.$$

$$g=f_1\dots f_n,$$

$$|\varphi (g)-\varphi (f_1)-\cdots -\varphi (f_n)|\le (n-1)(D(\varphi ))$$

$$\frac{|\varphi (g)|}{D(\varphi )+C_K}\le n.$$

$$\frac{|\varphi (g)|}{D(\varphi )+C_K}\le q_K(g).$$

$$\log \frac{|\varphi (g)|}{D(\varphi )+C_K}\le d([g],[K]).$$

$$\log n+\log \frac{|\varphi (g)|}{D(\varphi )+C_K}\le d([g^n],[K])\text{ for any }n.$$

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

$$g\mapsto {\int }_{x\in X_n(D^2)}\varphi (\gamma (g;x)){\Omega }^n,$$

$${\Gamma }_n(\varphi )(g)=\underset{p\to \infty }{\lim }\frac{1}{p}{\int }_{x\in X_n(D^2)}\varphi (\gamma (g^p;x)){\Omega }^n.$$

$${\phi }_r(x,y)=\left(\frac{x}{r},\frac{y}{r}\right).$$

$$s_{r}(f)(x,y)=\left\{\begin{array}[]{ll}\varphi_{r}\circ f\circ\varphi_{r}^{-1}(x,y)&\text{if }(x,y)\in D(r^{-1})\\ (x,y)&\text{if }(x,y)\notin D(r^{-1}).\end{array}\right.$$

$${\Gamma }_3\circ Q(i)(\varphi )(s_r(f))=\frac{1}{r^6}{\Gamma }_3\circ Q(i)(\varphi )(f)$$

$${\int }_{x\in X_3(D^2)}\varphi (\gamma (s_r(f);x)){\Omega }^3$$

$$+3{\int }_{x_1,x_2\in D(r^{-1}),x_3\notin D(r^{-1})}\varphi (\gamma (s_r(f);x)){\Omega }^3$$

$${\int }_{x_1,x_2,x_3\in D(r^{-1})}\varphi (\gamma (s_r(f);x)){\Omega }^3=\frac{1}{r^6}{\int }_{x\in X_3(D^2)}\varphi (\gamma (f;x)){\Omega }^3,$$

$$s_r(f)=(h_1{f}^{{\epsilon }_1}{h}_1^{-1})\dots (h_k{f}^{{\epsilon }_k}{h}_k^{-1}),$$

$$s_r^{n+1}(f)=(s_r^n(h_1)s_r^n(f{)}^{{\epsilon }_1}{s}_r^n(h_1{)}^{-1})\dots (s_r^n(h_k)s_r^n(f{)}^{{\epsilon }_k}{s}_r^n(h_k{)}^{-1})$$

$$\log m+\log \frac{|{\varphi }^{\prime }(g)|}{D({\varphi }^{\prime })+|{\varphi }^{\prime }(f)|}\le d([g^m],[f_n;n\ge 0])\text{ for any }m\in \mathbb{N},$$

Full text

1. Introduction

Suppose that $G$ is a simple group and $K\subseteq G$ is a subset. Here, we assume that $K$ contains non-trivial elements of $G$. Since the group $G$ is simple, any non-trivial element $g$ of $G$ can be written as a product of conjugates of elements of $K\cup K^{-1}$. We define for each $g\in G$ the number $q_{K}(g)$ by the minimal number of conjugates of elements of $K\cup K^{-1}$ whose product is equal to $g$. Here, for the identity element $e$, we define $q_{K}(e)=0$. The function $q_{K}\colon G\to{\mathbb{Z}}_{\geq 0}$ is obviously invariant under conjugations and defines a conjugation-invariant norm on $G$. Such a conjugation-invariant norm is called a conjugation-generated norm. In this paper, we mainly consider the case $K$ consists of a single non-trivial element.

Elements $f$ and $g$ of a group $G$ are symmetrized conjugate to each other if $f$ is conjugate to $g$ or $g^{-1}$. It is easy to see that symmetrized conjugacy is an equivalence relation. We denote by $[g]$ the symmetrized conjugacy class represented by $g\in G$. We define $\mathcal{M}({G})$ to be the set of non-trivial symmetrized conjugacy classes of elements of $G$. In [17], Tsuboi introduced a metric $d$ on $\mathcal{M}({G})$ defined by

[TABLE]

In fact, it is easy to see that the inequality

[TABLE]

holds for any $f,g,h\in G$ and thus the function $d\colon\mathcal{M}({G})\times\mathcal{M}({G})\to{\mathbb{R}}_{\geq 0}$ satisfies the triangle inequality. We are interested in this metric space $\mathcal{M}({G})$, which is an invariant of simple group.

In [12], Kodama studied the metric space $(\mathcal{M}({G}),d)$ for the case $G$ is the infinite alternating group $A_{\infty}$ and proved the following.

Theorem 1.1** (Kodama [12]).**

The metric space $(\mathcal{M}({A_{\infty}}),d)$ is quasi-isometric to the half line.

We define the $2$-disk $D^{2}$ and the standard area form $\Omega$ on $D^{2}$ to be

[TABLE]

respectively. Let ${\rm Diff}_{\Omega}^{\infty}(D^{2},\partial D^{2})$ be the group of $C^{\infty}$-diffeomorphisms of the $2$-disk $D^{2}$, which preserve $\Omega$ and are the identity on a neighborhood of the boundary. It is classically known that the group ${\rm Diff}_{\Omega}^{\infty}(D^{2},\partial D^{2})$ admits a homomorphism

[TABLE]

called the Calabi homomorphism. The Calabi homomorphism gives an abelianization of ${\rm Diff}_{\Omega}^{\infty}(D^{2},\partial D^{2})$ and its kernel ${\rm Ker}{\rm Cal}$ is simple [1]. In this paper, we study the metric space $(\mathcal{M}({G}),d)$ for the case $G={\rm Ker}{\rm Cal}$ and prove the following theorem.

Theorem 1.2**.**

For any non-trivial element $f\in{\rm Ker}{\rm Cal}$, there exist a sequence $\{f_{n}\}_{n\geq 0}$ contained in ${\rm Ker}{\rm Cal}$ with $f_{0}=f$, an element $g\in{\rm Ker}{\rm Cal}$ and positive constants $C_{1},C_{2},C_{3}$ which satisfy the following.

1. (i)

$d([f_{n}],[f_{m}])\geq C_{1}|n-m|$, 2. (ii)

$d([f_{n}],[f_{n+1}])\leq C_{2}$, 3. (iii)

$d([f_{n}],[g^{m}])\geq\log m+C_{3}$.

As a corollary, we obtain the following statement answering to a problem raised by Tsuboi [18, Problem4.4].

Theorem 1.3**.**

The metric space $(\mathcal{M}({{\rm Ker}{\rm Cal}}),d)$ is not quasi-isometric to the half line.

2. Quasi-morphisms

In this section, we prepare a notion of quasi-morphism, which is a useful tool to evaluate a lower bound for a conjugation-generated norm $q_{K}$ and prove Proposition 2.2. On quasi-morphisms and conjugation-generated norms, see [7] for more details.

Let $G$ be a group. A quasi-morphism on $G$ is a function $\phi\colon G\to{\mathbb{R}}$ such that there exists a constant $C\geq 0$ and $|\phi(gh)-\phi(g)-\phi(h)|\leq C$ for any $g,h\in G$. The real number

[TABLE]

is called the defect of $\phi$. A quasi-morphism $\phi$ on $G$ is homogeneous if $\phi(g^{p})=p\phi(g)$ for any $g\in G$ and any $p\in{\mathbb{Z}}$. For any quasi-morphism $\phi$ on an arbitrary group $G$, there exists a unique homogeneous quasi-morphism $\tilde{\phi}$ on $G$ such that $\tilde{\phi}-\phi$ is a bounded function on $G$ and $\tilde{\phi}$ is explicitly written as

[TABLE]

We denote by $Q(G)$ the ${\mathbb{R}}$-vector space consisting of homogeneous quasi-morphisms on $G$. Note that homogeneous quasi-morphisms are invariant under conjugations.

2.1. Conjugation-invariant norms and quasi-morphisms

Let $K$ be a subset of $G$. We define the vector subspace $Q(G,K)$ of $Q(G)$ by

[TABLE]

Note that this definition is different from that given in [7]. Suppose that $g\in G$ is written as

[TABLE]

where $f_{1},\dots,f_{n}$ are conjugates of elements of $K\cup K^{-1}$. Then for $\phi\in Q(G,K)$ the inequation

[TABLE]

holds. If we set $C_{K}=\sup_{h\in K}|\phi(h)|$, then we have

[TABLE]

This means that

[TABLE]

Denoting by $[K]$ the set of symmetrized conjugacy classes represented by the elements of $K$, we have the following lemma on the metric $d$ of $\mathcal{M}({G})$.

Lemma 2.1**.**

Let $\phi\in Q(G,K)$ and $g\in G$ such that $\phi(g)\neq 0$. Then

[TABLE]

In particular,

[TABLE]

A simple group $G$ is uniformly simple if the metric space $(\mathcal{M}({G}),d)$ is bounded. This is equivalent to saying that $(\mathcal{M}({G}),d)$ is quasi-isometric to a point. Since $Q(G,K)=Q(K)$ for any bounded set $K$, if the group $G$ admits a non-trivial quasi-morphism then $(\mathcal{M}({G}),d)$ is unbounded by Lemma 2.1 and thus $G$ is not uniformly simple.

2.2. Gambaudo-Ghys’ construction of quasi-morphisms on ${\rm Diff}_{\Omega}^{\infty}(D^{2},\partial D^{2})$

It is known that the vector space $Q({\rm Diff}_{\Omega}^{\infty}(D^{2},\partial D^{2}))$ is infinite-dimensional [8][9][10]. To prove Theorem 1.2, we use quasi-morphisms on ${\rm Ker}{\rm Cal}$ obtained by Brandenbursky generalizing Gambaudo-Ghys’ construction [5].

Let $X_{n}(D^{2})$ be the $n$-fold configuration space of $D^{2}$. Fix a base point $x^{0}=(x_{1}^{0},\dots,x_{n}^{0})\in X_{n}(D^{2})$. For any $g\in{\rm Diff}_{\Omega}^{\infty}(D^{2},\partial D^{2})$ and almost every $x=(x_{1},\dots,x_{n})\in X_{n}(D^{2})$, we set a loop $l(g;x)\colon[0,1]\to X_{n}(D^{2})$ by

[TABLE]

where $\{g_{t}\}_{t\in[0,1]}$ is a path in ${\rm Diff}_{\Omega}^{\infty}(D^{2},\partial D^{2})$ such that $g_{0}$ is the identity and $g_{1}=g$. Of course for some $x\in X_{n}(D^{2})$ the loop $l(g;x)$ may not be defined. However, for almost every $x$ the loop $l(g;x)$ is well-defined. We define the pure braid $\gamma(g;x)$ to be the homotopy class relative to the base point $x^{0}$ represented by the loop $l(g;x)$. Since the group of diffeomorphisms of $D^{2}$ is contractible [16] and is homotopy equivalence to ${\rm Diff}_{\Omega}^{\infty}(D^{2},\partial D^{2})$ [15], the pure braid $\gamma(g;x)$ is independent of the choice of the path $\{g_{t}\}$. Let $P_{n}(D^{2})$ be the pure braid group on $n$-strands. For a homogeneous quasi-morphism $\phi$ on $P_{n}(D^{2})$, if we consider the function

[TABLE]

then this function is well-defined [5][6] and is a quasi-morphism on ${\rm Diff}_{\Omega}^{\infty}(D^{2},\partial D^{2})$ since the diffeomorphism $g$ preserves $\Omega$. Thus we have the linear map $\Gamma_{n}\colon Q(P_{n}(D^{2}))\to Q({\rm Diff}_{\Omega}^{\infty}(D^{2},\partial D^{2}))$ defined by

[TABLE]

Let $B_{n}(D^{2})$ be the braid group on $n$ strands and $i\colon P_{n}(D^{2})\to B_{n}(D^{2})$ the natural inclusion. Then the linear map $Q(i)\colon Q(B_{n}(D^{2}))\to Q(P_{n}(D^{2}))$ is induced.

For $r>1$, we denote by $D(r^{-1})$ the small disk $\{(x,y)\in{\mathbb{R}};x^{2}+y^{2}\leq r^{-2}\}$ of radius $1/r$. Let $\varphi_{r}\colon D^{2}\to D(r^{-1})$ be the $C^{\infty}$-diffeomorphism defined by

[TABLE]

We define the homomorphism $s_{r}\colon{\rm Diff}_{\Omega}^{\infty}(D^{2},\partial D^{2})\to{\rm Diff}_{\Omega}^{\infty}(D^{2},\partial D^{2})$ by

[TABLE]

Note that if $f$ is in ${\rm Ker}{\rm Cal}$, then $s_{r}(f)$ is also.

Let $\sigma_{1}\in B_{3}(D^{2})$ be the braid on $3$ strands as indicated in Figure 1.

The following proposition is essentially introduced in [6, Lemma 3.11].

Proposition 2.2**.**

If $\phi\in Q(B_{3})$ satisfies $\phi(\sigma_{1})=0$, then

[TABLE]

for any $f\in{\rm Diff}_{\Omega}^{\infty}(D^{2},\partial D^{2})$ and any $r>1$.

Proof.

Let $x=(x_{1},x_{2},x_{3})$ be in $X_{3}(D^{2})$. For any $f\in{\rm Diff}_{\Omega}^{\infty}(D^{2},\partial D^{2})$ and any $r>1$, if two or three of $x_{1},x_{2},x_{3}$ are not in $D(r^{-1})$, then the pure braid $\gamma(s_{r}(f);x)$ is trivial. Hence we have

[TABLE]

for any $\phi\in Q(B_{3}(D^{2}))$.

If $x_{1},x_{2}\in D(r^{-1})$ and $x_{3}\not\in D(r^{-1})$, then the pure braid $\gamma(s_{r}(f);x)$ is a conjugate of a power of $\sigma_{1}$ and hence $\phi(\gamma(s_{r}(f);x))=0$. Since

[TABLE]

we have the desired equality. ∎

3. Proof of the main theorem

In this section, we prove the main theorem. Before starting the proof, we show the following lemma as a preliminary step.

Lemma 3.1**.**

For any $f\in{\rm Diff}_{\Omega}^{\infty}(D^{2},\partial D^{2})$ and $r>1$, the following holds.

1. (I)

$d([s_{r}^{m}(f)],[s_{r}^{n}(f)])\geq(2\log r)|m-n|$* .* 2. (II)

$d([s_{r}^{n}(f)],[s_{r}^{n+1}(f)])\leq d([f],[s_{r}(f)])$.

Proof.

Assume that $m<n$. Since the area of the support of $s_{r}^{m}(f)$ is just $r^{2(n-m)}$ times of that of $s_{r}^{n}(f)$, we have $q_{\{s_{r}^{n}(f)\}}(s_{r}^{m}(f))\geq r^{2(n-m)}$. This implies (I).

Suppose that $s_{r}(f)$ is written as a product

[TABLE]

where each $\varepsilon_{i}$ is $1$ or $-1$. Since the map $s_{r}\colon{\rm Diff}_{\Omega}^{\infty}(D^{2},\partial D^{2})\to{\rm Diff}_{\Omega}^{\infty}(D^{2},\partial D^{2})$ is a homomorphism, we have

[TABLE]

and thus $q_{\{s_{r}^{n}(f)\}}(s_{r}^{n+1}(f))\leq q_{\{f\}}(s_{r}(f))$. Similarly the inequality $q_{\{s_{r}^{n+1}(f)\}}(s_{r}^{n}(f))\leq q_{\{s_{r}(f)\}}(f)$ also holds. Hence we have (II). ∎

Proof of Theorem 1.2.

Fix $f\in{\rm Ker}{\rm Cal}$ and $r>1$. If we set $f_{n}=s_{r}^{n}(f)$, then the properties (i) and (ii) immediately follow from Lemma 3.1.

Since the vector space $Q(B_{n}(D^{2}))$ is infinite-dimensional for $n\geq 3$ [3], considering the linear combination it is guaranteed that there exists a non-trivial homogeneous quasi-morphism $\phi$ on $B_{3}$ satisfying $\phi(\sigma_{1})=0$. Since the composition of the linear maps $\Gamma_{n}\circ Q(i)\colon Q(B_{n}(D^{2}))\to Q({\rm Diff}_{\Omega}^{\infty}(D^{2},\partial D^{2}))$ is injective for $n\geq 3$ [10], its image $\Gamma_{3}\circ Q(i)(\phi)$ is also non-trivial. We denote it by $\phi^{\prime}$. By Proposition 2.2, $|\phi^{\prime}(f_{n})|\leq|\phi^{\prime}(f)|$ and thus $\phi^{\prime}$ is in $Q({\rm Diff}_{\Omega}^{\infty}(D^{2},\partial D^{2}),\{f_{n};n\geq 0\})$. Moreover, choose $g\in{\rm Ker}{\rm Cal}$ such that $\phi^{\prime}(g)\neq 0$. Then we have by Lemma 2.1

[TABLE]

which is the property (iii). ∎

Proof of Theorem 1.3.

If the metric spaces $\mathcal{M}({{\rm Ker}{\rm Cal}})$ and ${\mathbb{R}}_{\geq 0}$ are quasi-isometric, then there exists a quasi-isometric embedding $\Phi\colon\mathcal{M}({{\rm Ker}{\rm Cal}})\to{\mathbb{R}}_{\geq 0}$. By the property (iii), we have $\Phi([f])<\Phi([g^{m}])$ for sufficiently large $m\in{\mathbb{N}}$. By the property (i), there exists $n\in{\mathbb{N}}$ such that $\Phi([g^{m}])<\Phi([f_{n}])$. If we set $n_{m}=\min\{n\in{\mathbb{N}};\Phi([g^{m}])<\Phi([f_{n}])\}$, then $\Phi([f_{n_{m}}])-\Phi([g^{m}])$ is bounded independently on $m$ by the property (ii). However this contradicts the property (iii) since we can make $n_{m}$ arbitrarily large by taking larger $m$. ∎

Remark 3.2*.*

Let $M$ be a closed $C^{\infty}$-manifold and fix a symplectic form $\omega$ of $M$. Then the group ${\rm Ham}^{\infty}(M)$ of Hamiltonian diffeomorphisms of $M$ is a simple group [1].

Let $U$ be a closed ball in $M$. Taking the subgroup ${\rm Ham}^{\infty}(U)$ of ${\rm Ham}^{\infty}(M)$, consisting of diffeomorphisms supported by $U$, as in the case of $D^{2}$ we can consider the shrinking homomorphism $s_{r}\colon{\rm Ham}^{\infty}(U)\to{\rm Ham}^{\infty}(U)$ and construct a sequence $\{f_{n}\}$ in ${\rm Ham}^{\infty}(M)$ which satisfies the properties (i) and (ii) in Theorem 1.2. Hence if there exists a quasi-morphisms on ${\rm Ham}^{\infty}(M)$ whose restriction in ${\rm Ham}^{\infty}(U)$ have the property as Proposition 2.2, then Theorem 1.2 holds for ${\rm Ham}^{\infty}(M)$ and Theorem 1.3 for $\mathcal{M}({\rm Ham}^{\infty}(M))$.

When $M$ is a closed surface, we can construct quasi-morphisms on ${\rm Diff}_{\Omega}^{\infty}(M)_{0}$ by Gambaudo-Ghys’ way [4] and verify by an argument similar to the case of $D^{2}$ that there exists a quasi-morphism $\phi$ on ${\rm Ham}^{\infty}(M)$ satisfying $\phi(s_{r}(f))=r^{-6}\phi(f)$ for any $f\in{\rm Ham}^{\infty}(U)$.

When $M$ is the one point blow up of a closed symplectic $4$-manifold $(X,\omega_{X})$ such that $\omega_{X}$ and the first Chern class $c_{1}(X)$ vanish on $\pi_{2}(X)$, then ${\rm Ham}^{\infty}(M)$ admits a non-trivial quasi-morphism $\mu$, which is called a Calabi quasi-morphism [8][14]. If we take $U$ sufficiently small, then $\mu$ satisfies $\mu(s_{r}(f))=r^{-8}\mu(f)$ for any $f\in{\rm Ham}^{\infty}(U)$.

Remark 3.3*.*

Let ${\rm Ham}^{\infty}_{C}(D^{2n})$ and ${\rm Ham}^{\infty}_{C}({\mathbb{R}}^{2n})$ be the groups of Hamiltonian diffeomorphisms of $D^{2n}$ and ${\mathbb{R}}^{2n}$ respectively with respect to the standard symplectic form $\omega$. These groups admits the Calabi homomorphisms ${\rm Cal}\colon{\rm Ham}^{\infty}_{C}(D^{2n})\to{\mathbb{R}}$ and ${\rm Cal}_{{\mathbb{R}}}\colon{\rm Ham}^{\infty}_{C}({\mathbb{R}}^{2n})\to{\mathbb{R}}$ and their kernels ${\rm Ker}{\rm Cal}$ and ${\rm Ker}{\rm Cal}_{{\mathbb{R}}}$ are simple [1]. The group ${\rm Ham}^{\infty}_{C}(D^{2n})$ admits a quasi-morphism $\tau$, which is constructed by Barge and Ghys [2]. The quasi-morphism $\tau\in Q({\rm Ham}^{\infty}_{C}(D^{2n}))$ satisfies $\tau(s_{r}(f))=r^{-2n}(f)$.

Although the group ${\rm Ker}{\rm Cal}_{{\mathbb{R}}}$ does not admit non-trivial quasi-morphisms [13], Kawasaki constructed a homogeneous conjugation invariant function on ${\rm Ker}{\rm Cal}_{{\mathbb{R}}}$, which is called a partial quasi-morphism [11]. If we denote it by $\mu$, then the equation $\mu(s_{r}(f))={r^{-2n}}\mu(f)$ is satisfied.

Therefore a statement similar to Lemma 2.1 hold for $\tau$ and $\mu$. Hence Theorem 1.2 holds for ${\rm Ker}{\rm Cal}$ and ${\rm Ker}{\rm Cal}_{{\mathbb{R}}}$ and Theorem 1.3 for $\mathcal{M}({\rm Ker}{\rm Cal})$ and $\mathcal{M}({\rm Ker}{\rm Cal}_{{\mathbb{R}}})$.

Acknowledgments. The author wishes to express his gratitude to Jarek Kȩdra and Morimichi Kawasaki for reading the manuscript and several comments. The author is supported by JSPS Research Fellowships for Young Scientists (26$\cdot$110).