a)

A compact oriented surface $B$, a finite subset $B^{sing}\subset B$.

b)

A ${\bf Z}$-affine structure on $Y=B^{sm}=B\setminus B^{sing}$ with the standard singularities near each $b\in B^{sing}$.

c)

A set ${\cal L}$ of lines. With each line $l\in{\cal L}$ there is an associated continuous map $f_{l}:(0,+\infty)\to Y$. We assume that ${\cal L}$ is decomposed into a disjoint union of two subsets ${\cal L}={\cal L}_{in}\sqcup{\cal L}_{com}$. Lines belonging to ${\cal L}_{in}$ are called initial, while those in ${\cal L}_{com}$ are called composite. We assume that for any $l\in{\cal L}$ there exists a continuous extension $f_{l}:[0,+\infty)\to B$ such that $f_{l}(0)\in B^{sing}$ if $l\in{\cal L}_{in}$ and $f_{l}(0)\in Y=B^{sm}$ if $l\in{\cal L}_{com}$.

d)

A collection of covariantly constant nowhere vanishing integer-valued $1$-forms $\alpha_{l}\in\Gamma((0,+\infty),f_{l}^{\ast}((T^{\ast})^{\bf Z}),l\in{\cal L}$. We assume that for $l\in{\cal L}_{in}$ in the standard coordinates $(x,y)$ near singular point $f_{l}(0)$ we have: $f_{l}(t)=(0,t)$ or $f_{l}(t)=(0,-t)$ for all sufficiently small $t>0$, and $\alpha_{l}(t)=\pm f_{l}^{\ast}(dy)$.

e)

A map ${\cal L}\to{\cal L}\times{\cal L},\,\,\,l\mapsto(p_{left}(l),p_{right}(l))$ (the letter $p$ stands for “parent”: one can think about these lines as “generating $l$ in a collision”).

Notice that since the form $dy$ is invariant with respect to the monodromy, the condition in d) is coordinate-independent. The covector $\alpha_{l}(t)$ will be called a direction covector of $l$ at time $t$. It gives rise to a half-plane

$$P_{l,t}^{(0)}=\{v\in T_{f_{l}(t)}Y|\langle\alpha_{l}(t),v\rangle>0\}\,\,.$$

To every $l\in{\cal L}_{in}$ we assign a pair $\left(f_{l}(0),sgn\left(\alpha_{l}(0)\right)\right)\in B^{sing}\times\{\pm 1\}$, where $sgn(\alpha_{l}(0))$ is a choice of sign in $\pm f_{l}^{\ast}(dy)$ (see data d) in the previous subsection). In this way we obtain a map $r:{\cal L}_{in}\to B^{sing}\times\{\pm 1\}$.

Axiom 1.

Map $r$ is one-to-one.

Let $U\subset Y$ be a simply-connected domain, and line $l$ intersects $U$. Let $I\subset{\bf R}_{+}$ be an interval such that $f_{l}(I)\subset U$. Then there exists a covariantly constant closed non-zero $1$-form $\beta_{U}$ in $U$ (with constant integer coefficients), such that $f_{l}^{\ast}(\beta_{U})=\alpha_{l}$, when both sides are restricted to $I$.

Axiom 2.

For any $t_{1},t_{2}\in I$ one has

$$\int_{f_{l}(t_{1})}^{f_{l}(t_{2})}\beta_{U}=t_{2}-t_{1}\,\,.$$

Let $l_{1},l_{2}\in{\cal L}$, $t_{1},t_{2}>0$ satisfy the condition $f_{l_{1}}(t_{1})=f_{l_{2}}(t_{2})=x\in Y$. In this case we say that lines $l_{1}$ and $l_{2}$ have a collision at $x$ at the times $t_{1}$ and $t_{2}$ respectively.

Axiom 3.

Under the above assumptions there are only two possibilities:

3a) either $l_{1}=l_{2}$ and $t_{1}=t_{2}$, or

3b) covector $\alpha_{l_{1}}(t_{1})$ is not proportional to $\alpha_{l_{2}}(t_{2})$. Then we may assume that $\alpha_{l_{1}}(t_{1})\wedge\alpha_{l_{2}}(t_{2})>0$. Under these conditions we require that for any coprime positive integers $n_{1},n_{2}$ there exists a unique line $l\in{\cal L}$ such that $l_{1}=p_{left(l)},l_{2}=p_{right}(l)$, $f_{l}(0)=x$ and $\alpha_{l}(0)=n_{1}\alpha_{l_{1}}(t_{1})+n_{2}\alpha_{l_{2}}(t_{2})$.

In other words, $l_{1}$ and $l_{2}$ are “parents of $l$”, and the direction covector of $l$ at the intersection point is a primitive integral linear combination of those for $l_{1}$ and $l_{2}$ (see Figure 4).

Axiom 4.

For every line $l\in{\cal L}_{com}$ there exist $l_{1}$ and $l_{2}$ such that they satisfy the condition 3b).

Axiom 5.

For any $x\in Y$ there are no more than two pairs $(l,t)\in{\cal L}\times(0,+\infty)$ such that $x=f_{l}(t)$. In other words, there are no more than two lines intersecting at a point in $Y$.

Let $l_{1},l_{2},t_{1},t_{2},x$ mean the same as in the Axiom 3, and assume that $\alpha_{l_{1}}(t_{1})\wedge\alpha_{l_{2}}(t_{2})>0$. Let us consider the set ${\cal L}_{(x)}$ of germs of all $l\in{\cal L}_{com}$ starting at $x$ (i.e. such that $f_{l}(0)=x$).

Axiom 6.

For any finite subset ${\cal L}^{\prime}\subset{\cal L}_{(x)}$ there is an orientation preserving homeomorphism of a neighborhood of $x$ onto a neighborhood of $(0,0)\in{\bf R}^{2}$ such that:

6a) Germs of oriented curves which are images of $l_{1}$ and $l_{2}$ get transformed into the germs at $(0,0)$ of coordinate axes $(x,0)$ and $(0,y)$ respectively.

6b) Germ of the image of $l\in{\cal L}^{\prime}$ gets transformed into the germ of the ray $\{(n_{1}t,n_{2}t)\,|\,t>0\}$ where $\alpha_{l}(0)=n_{1}\alpha_{l_{1}}(t_{1})+n_{2}\alpha_{l_{2}}(t_{2})$.

Figure 5 illustrates this axiom.

Axiom 7.

Let $p_{i}$ denotes either $p_{left}$ or $p_{right}$. Then for any $l\in{\cal L}$ there exists $N\geq 1$ such that if the line $p_{1}(p_{2}(\dots p_{N}(l)\dots)$ is well-defined then it belongs to ${\cal L}_{in}$.

This axiom says that any composed line $l\in{\cal L}_{com}$ appears as a result of finitely many collisions. The tree of ancestors of a given line form a tree embedded in $B$, see Figure 6.

Here we offer a construction of the set of lines satisfying the above axioms.

Let us use the standard ${\bf R}^{2}$ as a model around each $b\in B^{sing}$ in order to fix a structure of smooth manifold on the whole surface $B$. Let $\widetilde{Y}$ denotes the covering of $Y$ such that the fiber over $y\in Y$ is $(T_{y}^{\ast}Y)^{{\bf Z}}\setminus\{0\}$.

Let us fix a generic smooth metric on $B$. By the pull-back it gives a metric on $\widetilde{Y}$. Notice that there is a canonical closed $1$-form $\beta$ on $\widetilde{Y}$ such that $\beta_{|(y,\mu)}=\mu$, where $y\in Y,\,\,\mu\in(T_{y}^{\ast}Y)^{{\bf Z}}$. Using the metric we obtain dual to $\beta$ gradient vector field $v$ on $\widetilde{Y}$.

For any $s\in B^{sing}$ and a choice of 1-form $\alpha(0)=\pm dy$ in local coordinates, we take the unique integral line of $v$ starting at $(s,\alpha(0))$. Set ${\cal L}_{in}$ will be the set of all lines obtained in this way. Each line $l\in{\cal L}_{in}$ carries a covariantly constant closed $1$-form $\alpha_{l}$. Using Axiom 2 as a definition, we obtain a canonical parametrization of each line by the time parameter $t$. Since the metric is generic, a line cannot return to a point in $B^{sing}$.

Then we proceed inductively. If two already constructed lines $l_{1},l_{2}\in{\cal L}$ meet at $x\in Y$ we produce a new integral line $l$ of $v$ with the direction covector satisfying the condition 3b) for any pair of coprime positive integers $n_{1},n_{2}$. In this way we construct a set of lines ${\cal L}$ satisfying all the axioms. The only non-trivial thing to check is that for each line values of the parameter $t$ are in one-to-one correspondence with the interval $(0,+\infty)$. In order to see this we observe that the length of each line is infinite. Indeed, an integral curve of $v$ cannot have a limiting point in $Y$ (since the flow generated by $v$ is smooth, and the lengths of tangent vectors are bounded from below because of the integrality of $1$-forms).

We conclude that there exists a set ${\cal L}$ of lines satisfying Axioms 1-7.

Here it will be convenient to work in local coordinates $(x,y)=(\log|\xi|,\log|\eta|)$ on $Y$.

Let $(x_{0},y_{0})\in{\bf R}^{2}$ be a point, $\alpha_{1},\alpha_{2}\in({\bf Z}^{2})^{\ast}$ be $1$-covectors such that $\alpha_{1}\wedge\alpha_{2}>0$. Denote by $V=V_{(x_{0},y_{0}),\alpha_{1},\alpha_{2}}$ the closed angle

$$\{(x,y)\in{\bf R}^{2}|\langle\alpha_{i},(x,y)-(x_{0},y_{0})\rangle\geq 0,i=1,2\,\}\,\,.$$

Let ${\cal O}(V)$ be a $K$-algebra consisting of series $f=\sum_{n,m\in{\bf Z}}c_{n,m}\xi^{n}\eta^{m}$, such that $c_{n,m}\in K$ and for all $(x,y)\in V$ we have:

1. 1.

   if $c_{n,m}\neq 0$ then $\langle(n,m),(x,y)-(x_{0},y_{0})\rangle\leq 0$, where we identified $(n,m)\in{\bf Z}^{2}$ with a covector in $(T_{p}^{\ast}Y)^{{\bf Z}}$;

2. 2.

   $\log|c_{n,m}|+nx+my\to-\infty$ as long as $|n|+|m|\to+\infty$.

Algebra ${\cal O}(V)$ is a Poisson algebra with respect to the bracket $\{\xi,\eta\}=\xi\eta$.

For an integer covector $\mu=adx+bdy\in({\bf Z}^{2})^{*}$ we denote by $R_{\mu}$ the monomial $\xi^{a}\eta^{b}$.

Let us consider a pro-nilpotent Lie algebra ${\bf g}:={\bf g}_{\alpha_{1},\alpha_{2},V}\subset{\cal O}(V)$ consisting of series

$$f=\sum_{n_{1},n_{2}\geq 0,n_{1}+n_{2}>0}c_{n_{1},n_{2}}R_{\alpha_{1}}^{-n_{1}}R_{\alpha_{2}}^{-n_{2}}$$

satisfying the condition

$$\log|c_{n,m}|-n_{1}\langle\alpha_{1},(x,y)\rangle-n_{2}\langle\alpha_{2},(x,y)\rangle\leq 0\,\,\,\,\forall\,(x,y)\in V\,\,.$$

The latter condition is equivalent to $\log|c_{n,m}|-\langle n_{1}\alpha_{1}+n_{2}\alpha_{2},(x_{0},y_{0})\rangle\leq 0$.

Lie algebra ${\bf g}$ admits a filtration by Lie subalgebras ${\bf g}^{\geq k},k\in{\bf Z},k\geq 1$, ${\bf g}={\bf g}^{\geq 1}$, such that ${\bf g}^{\geq k}$ consists of the above series which satisfy the condition $n_{1}+n_{2}\geq k$. Clearly $[{\bf g}^{\geq k_{1}},{\bf g}^{\geq k_{2}}]\subset{\bf g}^{\geq k_{1}+k_{2}}$, and ${\bf g}=\varprojlim_{k\to+\infty}{\bf g}/{\bf g}^{\geq k}$.

Thus, ${\bf g}$ is a topological complete pro-nilpotent Lie algebra over $K$. We denote by $G$ the corresponding pro-nilpotent Lie group $\exp({\bf g})$. It inherits the filtration by normal subgroups $G^{\geq k}$ obtained from the corresponding Lie algebras.

For each $\lambda\in[0,+\infty]_{{\bf Q}}:={\bf Q}_{\geq 0}\cup\infty$ we define a Lie subalgebra

$${\bf g}_{\lambda}=\left\{\sum_{n_{1},n_{2}}c_{m,n}R_{\alpha_{1}}^{-n_{1}}R_{\alpha_{2}}^{-n_{2}}\in{\bf g}\,|\,\,c_{n_{1},n_{2}}\in K,\,\,{n_{2}\over{n_{1}}}=\lambda\,\right\}\,\,.$$

Each ${\bf g}_{\lambda}$ is an abelian Lie algebra. It carries the induced filtration by Lie algebras ${\bf g}_{\lambda}^{\geq k}={\bf g}_{\lambda}\cap{\bf g}^{\geq k}$. Denote by $G_{\lambda}=\exp({\bf g}_{\lambda})$ the corresponding pro-nilpotent group.

Proof. Indeed, for the monomial $R_{\alpha_{1}}^{-n_{1}}R_{\alpha_{2}}^{-n_{2}}\in{\bf g}_{\lambda}$ which maps non-trivially to the quotient ${\bf g}_{\lambda}/{\bf g}^{\geq k}_{\lambda}$ we have: $n_{1}+n_{2}\leq k,n_{1}/n_{2}=\lambda$, where $n_{1},n_{2}$ are non-negative integers. There are finitely many such non-negative integers $n_{1}$ and $n_{2}$. $\blacksquare$

It follows from the Lemma that we have a natural isomorphism of vector spaces $\prod_{\lambda\in[0,+\infty]_{{\bf Q}}}{\bf g}_{\lambda}/{\bf g}_{\lambda}^{\geq k}\to{\bf g}/{\bf g}^{\geq k}$, hence the map

$$(f_{\lambda})_{\lambda\in[0,+\infty]_{{\bf Q}}}\mapsto\sum_{\lambda}f_{\lambda}=\sum_{i=1}^{N_{i}}f_{\lambda_{i}},\,\,\,\,\mbox{ where }f_{\lambda}\in{\bf g}_{\lambda}/{\bf g}_{\lambda}^{\geq k}\,\,\,\forall\lambda\in[0,+\infty]_{{\bf Q}}$$

is well-defined and gives rise (after taking the projective limit as $k\to+\infty$) to the isomorphism ${\bf g}\simeq\prod_{\lambda\in[0,+\infty]_{{\bf Q}}}{\bf g}_{\lambda}$.

In a similar way we define the map $\prod_{\to}:\prod_{\lambda\in[0,+\infty]_{{\bf Q}}}G_{\lambda}\to G$, the product is taken with respect to the natural order on ${\bf Q}$. Namely, for any $k\geq 1$ we define

$${\textstyle\prod_{\to}^{(k)}}:\prod_{i=1}^{N_{k}}G_{\lambda_{i}}/G_{\lambda_{i}}^{\geq k}\to G/G^{\geq k}\,,\,(g_{1},\dots,g_{N_{k}})\mapsto g_{1}\dots g_{N_{k}},\,\mbox{ for }g_{i}\in G_{\lambda_{i}}/G_{\lambda_{i}}^{\geq k}$$

and then set $\prod_{\to}:=\varprojlim_{k}\prod_{\to}^{(k)}\,$.

Proof. Let $k\geq 1$ be an integer. We claim that $\prod_{\to}^{(k)}$ is a bijection of sets (this implies the proposition by taking the projective limit as $k\to+\infty$). We will prove the bijection by induction in $k$ . Case $k=1$ is obvious because all the groups under considerations are trivial.

We would like to prove that $\prod_{\to}^{(k+1)}$ is a bijection assuming that $\prod_{\to}^{(k)}$ is a bijection. Let $h$ be an element of $G/G^{\geq k+1}$ and $\overline{h}$ its image in $G/G^{\geq k}$. By the induction assumption there exist unique $\overline{h}_{i}\in G_{\lambda_{i}}/G_{\lambda_{i}}^{\geq k+1},1\leq i\leq N_{k+1}$ such that $\overline{h}_{1}\dots\overline{h}_{N_{k+1}}=\overline{h}$. Let $h_{i},1\leq i\leq N_{k+1}$ be any liftings of $\overline{h}_{i}$ to $G_{i}/G_{i}^{\geq k}$. Then $h_{1}\dots h_{N_{k+1}}=h\pmod{G^{\geq k}}$, hence $c:=h_{1}\dots h_{N_{k+1}}h^{-1}$ belongs to $G^{\geq k}/G^{\geq k+1}\subset Center(G/G^{\geq k+1})$. The last inclusion holds because $[{\bf g},{\bf g}^{\geq k}]=[{\bf g}^{\geq 1},{\bf g}^{\geq k}]\subset{\bf g}^{\geq k+1}$.

Next we observe that the isomorphism of abelian Lie algebras

$$\bigoplus_{1\leq i\leq{N_{k+1}}}{\bf g}_{\lambda_{i}}^{\geq k}/{\bf g}_{\lambda_{i}}^{\geq k+1}\simeq{\bf g}^{\geq k}/{\bf g}^{\geq k+1}$$

implies an isomorphism of the corresponding abelian groups

$$\prod_{1\leq i\leq{N_{k+1}}}G_{i}^{\geq k}/G_{i}^{\geq k+1}\simeq G^{\geq k}/G^{\geq k+1}\,\,.$$

Hence we can write uniquely $c=c_{1}\dots c_{N_{k+1}}$, where $c_{i}\in G^{\geq k}/G^{\geq k+1}\subset Center(G/G^{\geq k+1})$. It follows that $\prod_{\to}^{(k+1)}\left((h_{i}c_{i}^{-1})\right)=h$. Also it is now clear that this decomposition of $h$ is unique. This concludes the proof. $\blacksquare$

For $l\in{\cal L}$ we will define an order function

$$ord_{l}\in\Gamma((0,+\infty),f_{l}^{\ast}(Aff_{{\bf Z},Y}))$$

(its meaning will become clear later) by the following inductive procedure:

1. 1.

   Let $l\in{\cal L}_{in}$ and $t>0$ be sufficiently small. Then in the standard affine coordinates near $s=f_{l}(0)$ one has $\alpha_{l}=\pm f_{l}^{\ast}(dy)$. We define $ord_{l}=\pm f_{l}^{\ast}(y)$. Then $d(ord_{l})=\alpha_{l}$, and we can extend uniquely $ord_{l}$ for all $t\in(0,+\infty)$.

2. 2.

   Let $l\in{\cal L}_{com}$ and $l_{1},l_{2}$ be parents of $l$. In the notation of Axiom 3 we have $f_{l_{1}}(t_{1})=f_{l_{2}}(t_{2})=f_{l}(0)$ and $\alpha_{l}(0)=n_{1}\alpha_{l_{1}}(t_{1})+n_{2}\alpha_{l_{2}}(t_{2})$. Then we define $ord_{l}(0):=n_{1}ord_{l_{1}}(t_{1})+n_{2}ord_{l_{2}}(t_{2})$. Again, using the condition $d(ord_{l})=\alpha_{l}$ and the knowledge of $ord_{l}(0)$ we can extend $ord_{l}$ for $t>0$.

Notice that $ord_{l}(t)$ can be thought of as affine function on the tangent space $T_{f_{l}(t)}Y$ (in the induced integral affine structure). In particular, we have a half-plane $P_{l,t}\subset T_{f_{l}(t)}Y$ defined by the inequality $ord_{l}(t)>0$. The family of half-planes $P_{l,t}$ is covariantly constant with respect to $\nabla^{aff}$.

Each half-plane $P_{l,t}$ contains $0\in T_{f_{l}(t)}Y$ strictly in its interior. Recall that at the end of Section 9.1 we defined another half-plane $P_{l,t}^{(0)}\subset T_{f_{l}(t)}Y$. It is easy to see that $P_{l,t}^{(0)}$ is the half-plane parallel to $P_{l,t}$ such that $0\in T_{f_{l}(t)}Y$ is on the boundary of $P_{l,t}^{(0)}$.