Read-back: definitions for piecewise projective homeomorphisms of the projective line, coamenability, inner amenability and amenable measured relations

This item is a bundle of definitions. They share the following vocabulary.

The projective line. Throughout, $\mathbb{P}^1$ denotes $\mathbb{R}\cup\{\infty\}$, the one-point compactification of $\mathbb{R}$ with its usual topology: a set $U\subseteq\mathbb{P}^1$ is open exactly when $U\cap\mathbb{R}$ is open in $\mathbb{R}$ and, if $\infty\in U$, the complement $\mathbb{R}\setminus U$ is compact. So the neighbourhoods of a real point $x$ are the sets containing an open interval around $x$, and the neighbourhoods of $\infty$ are the sets containing $\{\infty\}\cup(\mathbb{R}\setminus K)$ for some compact $K\subseteq\mathbb{R}$.

The homeomorphism group. $\mathrm{Homeo}(\mathbb{P}^1)$ denotes the group of all homeomorphisms $\mathbb{P}^1\to\mathbb{P}^1$, with product $(f g)(x)=f(g(x))$ (composition, $g$ applied first), identity the identity map, and inverse the inverse map. "The subgroup generated by a set $S$" means the smallest subgroup of $\mathrm{Homeo}(\mathbb{P}^1)$ containing $S$. $H_1\cap H_2$ is the intersection of subgroups.

Subrings. Throughout, $A$ is an arbitrary subring of $\mathbb{R}$ (containing $1$). Two particular subrings appear: the whole of $\mathbb{R}$, and the smallest subring, which is $\mathbb{Z}$ (the set of real numbers of the form $n\cdot 1$, $n\in\mathbb{Z}$). $\mathrm{SL}_2(A)$ is the group of $2\times 2$ matrices with entries in $A$ and determinant $1$.

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1. The embedding $\mathrm{SL}_2(A)\to\mathrm{GL}_2(\mathbb{R})$

For a subring $A\subseteq\mathbb{R}$, this is the group homomorphism $\mathrm{SL}_2(A)\to\mathrm{GL}_2(\mathbb{R})$ that sends a matrix with entries in $A$ to the same matrix regarded as a real matrix (entries included into $\mathbb{R}$). It is injective.

2. The Möbius action of $\mathrm{SL}_2(A)$ on $\mathbb{P}^1$

For a subring $A$ (left implicit, determined by $g$), a matrix

$$g=\begin{pmatrix}a&b\\ c&d\end{pmatrix}\in\mathrm{SL}_2(A)$$

and a point $x\in\mathbb{P}^1$, the point $g\cdot x\in\mathbb{P}^1$ is obtained by regarding $g$ as a real matrix and applying the standard action of $\mathrm{GL}_2(\mathbb{R})$ on $\mathbb{P}^1$ (via the identification of $\mathbb{P}^1$ with the lines in $\mathbb{R}^2$: a real $t$ corresponds to the line through $(t,1)$ and $\infty$ to the line through $(1,0)$; $g$ acts on column vectors). Explicitly, for real $t$,

$$g\cdot t=\begin{cases}\infty & \text{if } ct+d=0,\\[2pt] \dfrac{at+b}{ct+d} & \text{otherwise,}\end{cases} \qquad g\cdot\infty=\begin{cases}\infty & \text{if } c=0,\\[2pt] a/c & \text{otherwise.}\end{cases}$$

In particular $g$ and $-g$ act identically.

3. Hyperbolic elements

A matrix $g=\begin{pmatrix}a&b\\ c&d\end{pmatrix}\in\mathrm{SL}_2(A)$ is called hyperbolic when its trace, computed in $A$ and regarded as a real number, has absolute value strictly greater than $2$:

$$|a+d|>2 .$$

4. The set $P_A$ of hyperbolic fixed points

For a subring $A$,

$$P_A=\{\,p\in\mathbb{P}^1 : \text{there is a hyperbolic } g\in\mathrm{SL}_2(A)\text{ with } g\cdot p=p\,\}.$$

The point $\infty$ is allowed.

5. Piecewise projective on a set of breakpoints

Given a subring $A$, a set $E\subseteq\mathbb{P}^1$ and a homeomorphism $f$ of $\mathbb{P}^1$, $f$ is piecewise $A$-projective with breakpoints in $E$ when there exists a finite set $B\subseteq\mathbb{P}^1$ with $B\subseteq E$ such that for every point $x\in\mathbb{P}^1\setminus B$ there exists a matrix $g\in\mathrm{SL}_2(A)$ (which may depend on $x$) and a neighbourhood $U$ of $x$ in $\mathbb{P}^1$ with

$$f(y)=g\cdot y\quad\text{for all } y\in U.$$

Nothing is required at the points of $B$ themselves. $B$ may be empty (then $f$ is locally Möbius at every point). The condition allows $x=\infty$, in which case $U$ is a neighbourhood of $\infty$ as described above.

6. Piecewise projective (breakpoints in $P_A$)

A homeomorphism $f$ of $\mathbb{P}^1$ is piecewise $A$-projective when it is piecewise $A$-projective with breakpoints in $P_A$ (definition 5 with $E=P_A$ from definition 4): some finite set $B\subseteq P_A$ exists such that every $x\notin B$ has a neighbourhood on which $f$ agrees with the Möbius action of a single element of $\mathrm{SL}_2(A)$.

7. The group $G_{pp}$

$G_{pp}$ is the subgroup of $\mathrm{Homeo}(\mathbb{P}^1)$ generated by all homeomorphisms $f$ that are piecewise $\mathbb{R}$-projective with breakpoints anywhere, i.e. definition 5 with $A=\mathbb{R}$ and $E=\mathbb{P}^1$: there is a finite set $B\subseteq\mathbb{P}^1$ such that every $x\notin B$ has a neighbourhood on which $f$ coincides with $y\mapsto g\cdot y$ for some $g\in\mathrm{SL}_2(\mathbb{R})$.

8. The group $G(A)$

For a subring $A$, $G(A)$ is the subgroup of $\mathrm{Homeo}(\mathbb{P}^1)$ generated by the set of homeomorphisms $f$ such that both

• $f\in G_{pp}$, and
• $f$ is piecewise $A$-projective in the sense of definition 6 (finitely many breakpoints, all in $P_A$; locally given by elements of $\mathrm{SL}_2(A)$ elsewhere).

(Any $f$ satisfying the second condition also satisfies the first, since an element of $\mathrm{SL}_2(A)$ is also an element of $\mathrm{SL}_2(\mathbb{R})$ acting by the same map, and $P_A\subseteq\mathbb{P}^1$; so the generating set is exactly the set of piecewise $A$-projective homeomorphisms.)

9. The stabiliser of $\infty$

$\mathrm{Fix}(\infty)$ is the subgroup of $\mathrm{Homeo}(\mathbb{P}^1)$ consisting of all homeomorphisms $f$ with $f(\infty)=\infty$.

10. The group $H(A)$

For a subring $A$,

$$H(A)=G(A)\cap\mathrm{Fix}(\infty),$$

the elements of $G(A)$ (definition 8) that fix $\infty$.

11. The group $H_{pp}$

$$H_{pp}=G_{pp}\cap\mathrm{Fix}(\infty),$$

the elements of $G_{pp}$ (definition 7) that fix $\infty$.

12. The rational points

$$\mathbb{P}^1(\mathbb{Q})=\{\infty\}\cup\{\,q : q\in\mathbb{Q}\,\}\subseteq\mathbb{P}^1,$$

the point at infinity together with all rational numbers (viewed as real numbers, viewed as points of $\mathbb{P}^1$).

13. The group $G_{\mathbb{Q}}$

$G_{\mathbb{Q}}$ is the subgroup of $\mathrm{Homeo}(\mathbb{P}^1)$ generated by the set of homeomorphisms $f$ such that both

• $f\in G_{pp}$, and
• $f$ is piecewise $\mathbb{Z}$-projective with breakpoints in $\mathbb{P}^1(\mathbb{Q})$ (definition 5 with $A=\mathbb{Z}$, the smallest subring of $\mathbb{R}$, and $E=\mathbb{P}^1(\mathbb{Q})$): there is a finite set $B$ of points of $\mathbb{P}^1(\mathbb{Q})$ such that every $x\notin B$ has a neighbourhood on which $f$ coincides with $y\mapsto g\cdot y$ for some $g\in\mathrm{SL}_2(\mathbb{Z})$.

Note that the breakpoint set here is $\mathbb{P}^1(\mathbb{Q})$, not the set $P_{\mathbb{Z}}$ of definition 4. (As in definition 8, the second condition implies the first.)

14. The group $H_{\mathbb{Q}}$

$$H_{\mathbb{Q}}=G_{\mathbb{Q}}\cap\mathrm{Fix}(\infty),$$

the elements of $G_{\mathbb{Q}}$ that fix $\infty$.

15. Pointwise fixator of a set inside a subgroup

For a subgroup $J\le\mathrm{Homeo}(\mathbb{P}^1)$ and a set $E\subseteq\mathbb{P}^1$, this is the subgroup of $J$ (a subgroup of the group $J$ itself, not of $\mathrm{Homeo}(\mathbb{P}^1)$)

$$J_E=\{\,h\in J : h(x)=x\text{ for every } x\in E\,\}.$$

If $E$ is empty this is all of $J$.

16. Coamenable subgroup

Let $J$ be any group and $K$ a subgroup of $J$. Write $J/K$ for the set of left cosets $aK$ ($a\in J$), on which $J$ acts by left multiplication, $g\cdot(aK)=(ga)K$; for a set $S\subseteq J/K$ put $g\cdot S=\{g\cdot s : s\in S\}$. $K$ is coamenable in $J$ when there exists a function

$$m:\{\text{all subsets of } J/K\}\to[0,\infty]$$

such that

1. $m(\varnothing)=0$, and $m(S\cup T)=m(S)+m(T)$ whenever $S,T\subseteq J/K$ are disjoint (finite additivity, on arbitrary subsets);
2. $m(J/K)=1$;
3. $m(g\cdot S)=m(S)$ for every $g\in J$ and every $S\subseteq J/K$.

(From 1 and 2, $m$ is monotone and takes values in $[0,1]$.) No hypothesis on $J$ or $K$ beyond being a group and a subgroup; in particular $J/K$ may be a single point, in which case the condition holds.

17. Inner amenable group

A group $J$ is inner amenable when there exists a function

$$m:\{\text{all subsets of } J\}\to[0,\infty]$$

such that

1. $m(\varnothing)=0$ and $m(S\cup T)=m(S)+m(T)$ for all disjoint $S,T\subseteq J$;
2. $m(J)=1$;
3. $m(\{1\})=0$, where $1$ is the identity of $J$;
4. $m(gSg^{-1})=m(S)$ for every $g\in J$ and every $S\subseteq J$, where $gSg^{-1}=\{gxg^{-1}:x\in S\}$.

If $J$ is the trivial group, conditions 2 and 3 contradict each other, so the trivial group is not inner amenable under this definition.

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The remaining definitions concern a set $X$ equipped with a $\sigma$-algebra (a measurable space), in an arbitrary universe; $X\times X$ carries the product $\sigma$-algebra and $\mathbb{R}$ its Borel $\sigma$-algebra. A measure $\mu$ on $X$ is a countably additive measure on this $\sigma$-algebra; $\mu(S)$ for an arbitrary (possibly non-measurable) set $S\subseteq X$ means its outer measure, $\inf\{\mu(T): T\supseteq S \text{ measurable}\}$. "$\mu$-a.e." means outside a $\mu$-null set. $R\subseteq X\times X$ is an arbitrary subset (not assumed measurable, nor an equivalence relation). $\pi_1:X\times X\to X$ is the first-coordinate projection.

18. $R$-null sets

For a measure $\mu$ on $X$, a set $R\subseteq X\times X$ and a set $S\subseteq X\times X$, $S$ is $R$-null (for $\mu$) when

$$\mu\bigl(\pi_1(S\cap R)\bigr)=0,$$

i.e. the set of first coordinates of points of $R$ lying in $S$ has $\mu$-(outer) measure zero.

19. Bounded measurable on $R$

For $R\subseteq X\times X$ and $f:X\times X\to\mathbb{R}$, $f$ is bounded measurable on $R$ when

• $f$ is measurable on all of $X\times X$ (product $\sigma$-algebra to Borel sets), and
• there is a real constant $C$ with $|f(p)|\le C$ for every $p\in R$.

Boundedness is required only on $R$; measurability is required everywhere.

20. Partial transformations of $R$

For $R\subseteq X\times X$, a partial transformation of $R$ consists of

• two measurable sets $D,C'\subseteq X$ (the domain and codomain; either may be empty),
• a bijection $\varphi:D\to C'$ that is measurable with measurable inverse, where $D$ and $C'$ carry the subspace $\sigma$-algebras (a measurable isomorphism),

subject to the condition that its graph lies in $R$: $(x,\varphi(x))\in R$ for every $x\in D$.

Associated to a partial transformation $\varphi:D\to C'$ are two shift operations.

• For $f:X\times X\to\mathbb{R}$, the function $\varphi_*f:X\times X\to\mathbb{R}$ is

$$\varphi_*f(y,z)=\begin{cases} f\bigl(\varphi^{-1}(y),\,z\bigr) & \text{if } y\in C',\\ 0 & \text{otherwise.}\end{cases}$$

(It is defined for every pair $(y,z)$, whether or not $(\varphi^{-1}(y),z)$ or $(y,z)$ lies in $R$.)

• For $F:X\to\mathbb{R}$, the function $\varphi_*F:X\to\mathbb{R}$ is

$$\varphi_*F(y)=\begin{cases} F\bigl(\varphi^{-1}(y)\bigr) & \text{if } y\in C',\\ 0 & \text{otherwise.}\end{cases}$$

21. Left-invariant mean on $R$

Let $\mu$ be a measure on $X$, $R\subseteq X\times X$, and let $\mathcal{P}$ be any map sending every function $f:X\times X\to\mathbb{R}$ (not only the bounded measurable ones) to a function $\mathcal{P}f:X\to\mathbb{R}$. $\mathcal{P}$ is a left-invariant mean on $R$ (for $\mu$) when all of the following hold. In each item, $f,g$ range over functions that are bounded measurable on $R$ (definition 19), except where said otherwise; all equalities and inequalities between functions on $X$ are required only $\mu$-almost everywhere.

1. Measurability. For every bounded measurable $f$ on $R$, $\mathcal{P}f$ is $\mu$-almost-everywhere measurable (it agrees $\mu$-a.e. with a measurable function $X\to\mathbb{R}$).
2. Null-set invariance. If $f,g$ are bounded measurable on $R$ and the set $\{p\in X\times X: f(p)\neq g(p)\}$ is $R$-null (definition 18) — that is, $\mu\bigl(\{x : \exists z,\ (x,z)\in R,\ f(x,z)\neq g(x,z)\}\bigr)=0$ — then $\mathcal{P}f=\mathcal{P}g$ $\mu$-a.e.
3. Additivity. $\mathcal{P}(f+g)=\mathcal{P}f+\mathcal{P}g$ $\mu$-a.e.
4. Homogeneity. For every real $c$, $\mathcal{P}(cf)=c\,\mathcal{P}f$ $\mu$-a.e.
5. Positivity. If $f$ is bounded measurable on $R$ and $f(p)\ge 0$ for every $p\in R$ (no condition off $R$), then $\mathcal{P}f\ge 0$ $\mu$-a.e.
6. Normalisation. $\mathcal{P}\mathbf{1}=\mathbf{1}$ $\mu$-a.e., where $\mathbf{1}$ denotes the constant function $1$ (on $X\times X$, respectively on $X$).
7. Invariance. For every partial transformation $\varphi$ of $R$ (definition 20) and every $f$ bounded measurable on $R$,

$$\mathcal{P}(\varphi_*f)=\varphi_*(\mathcal{P}f)\quad\mu\text{-a.e.},$$

with both shifts as in definition 20. (No assumption is made that $\varphi_*f$ is itself bounded measurable on $R$.)

The "a.e." set may depend on $f$, $g$, $c$ and $\varphi$. On functions that are not bounded measurable on $R$, $\mathcal{P}$ is constrained only through items 6 and 7 as stated. If $\mu$ is the zero measure, every "a.e." condition holds automatically.

22. Amenable measured relation

For a measure $\mu$ on $X$ and $R\subseteq X\times X$, $R$ is amenable (for $\mu$) when there exists a left-invariant mean on $R$ for $\mu$ in the sense of definition 21, i.e. some map $\mathcal{P}$ from functions $X\times X\to\mathbb{R}$ to functions $X\to\mathbb{R}$ satisfying items 1–7.

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23. The $\sigma$-algebra on $\mathbb{P}^1$

$\mathbb{P}^1$ is given the Borel $\sigma$-algebra of its topology (the $\sigma$-algebra generated by the open sets described at the top). This is declared as the standard $\sigma$-algebra on $\mathbb{P}^1$ from this point on (globally, not only inside this item), and $\mathbb{P}^1$ is recorded as a Borel space with it.

24. The measure on $\mathbb{P}^1$

The measure $\lambda_{\mathbb{P}^1}$ on $\mathbb{P}^1$ (with the Borel $\sigma$-algebra of definition 23) is the pushforward of Lebesgue measure $\lambda$ on $\mathbb{R}$ under the inclusion $\iota:\mathbb{R}\to\mathbb{P}^1$:

$$\lambda_{\mathbb{P}^1}(S)=\lambda\bigl(\iota^{-1}(S)\bigr)=\lambda(S\cap\mathbb{R})\qquad\text{for Borel } S\subseteq\mathbb{P}^1 .$$

(The inclusion is continuous, hence measurable, so the pushforward is the genuine one.) In particular $\lambda_{\mathbb{P}^1}(\{\infty\})=0$ and $\lambda_{\mathbb{P}^1}(\mathbb{P}^1)=\infty$: it is an infinite measure, not a probability measure.