1. Introduction
Dicks and Sunic gave an elegant way of totally ordering the vertex set of a directed tree [Reference Dicks and Šunić9]. They applied this to give a simple proof of Vinogradov’s result that free groups, and more generally, free products of left-orderable groups are left-orderable. The purpose of this text is to describe a cyclically ordered counterpart.
Our basic observation is that:
Lemma 1.1. Let 
$T = (V,E)$ be a tree. Suppose there is a cyclic order on 
$\text{link}(v)$ for each 
$v \in V$. Then there is an induced cyclic order on the directed edges of T.
Using this natural cyclic order, we examine graphs of groups and obtain:
Theorem 1.2. Let G split as a graph of groups with left-cyclically ordered vertex groups and convex left-ordered edge groups. Then G is left-cyclically ordered in a manner compatible with its vertex and edge groups.
This generalizes the result of Baik and Samperton that free products of left-cyclically ordered groups are left-cyclically ordered [Reference Baik and Samperton2]. Another recent study probing more deeply than our own, was given by Clay and Ghaswala who characterized when an amalgam of cyclically ordered groups is cyclically ordered [Reference Clay and Ghaswala5]. The approach of Clay and Ghaswala specializes to give a proof of the amalgamated product case of our result. Moreover, it was pointed out to us that Calegari suggested a similar approach to cyclically ordering the boundary of the Bass–Serre tree of an amalgamated product [Reference Calegari4, Ex 2.116], from which one can sometimes deduce a cyclic ordering on the amalgam after some additional care and hypotheses.
There has been increased activity in the study of cyclically ordered groups, which are a bit more general than ordered groups. Some perspective on the relationship between them is given by the intriguing characterization that G is left-ordered if and only if 
$G \times {\mathbb{Z}}_n$ is left cyclically ordered for each n [Reference Bell, Clay and Ghaswala1]. Finally, we refer to [Reference Ghys10] and [Reference Calegari3] for surveys on cyclically ordered groups.
2. Cyclic orders
Definition 2.1. (Cyclic order) A cyclic order on a set A is a function
$\Theta : A \times A \times A \rightarrow \{-1, 0, 1\}$ satisfying the following conditions:
• Non-degeneracy:
$\Theta(x,y,z) = \pm1$ if and only if $x,y,z$ are pairwise distinct.• Cyclicity: If
$\Theta(x,y,z) = 1$, then $\Theta(z,x,y) = 1$.• Asymmetry:
$\Theta(x,y,z) = -\Theta(y,x,z)$.• Transitivity: If
$\Theta(x,y,z) = 1$ and $\Theta(x,z,w) = 1$, then $\Theta(x,y,w) = 1$.
We write 
$[x,y,z]$ whenever 
$\Theta(x,y,z) = 1$.
Definition 2.2. A strict total order is a binary relation 
$\prec$ on a set X satisfying the following conditions for all 
$x,y,z \in X$:
• Irreflexivity:
$x \not \prec x$ for all $x \in X$.• Comparability: if x ≠ y then
$x \prec y$ or $y \prec x$.• Transitivity: if
$x \prec y$ and $y \prec z$ then $x \prec z$.
The associated total order is denoted by 
$x \preceq y$ which means 
$x \prec y$ or x = y. We refer to 
$(X, \preceq)$ as a totally ordered set, and 
$(X, \prec)$ as a strict-totally ordered set.
Remark 2.3. For a strict-totally ordered set 
$(X, \prec)$, an associated cyclic order on X is defined by: 
$[x,y,z]$ holds provided 
$x \prec y \prec z$ or 
$y \prec z \prec x$ or 
$z \prec x \prec y$.
Remark 2.4. Consider 
$[0, 2\pi)$ with the usual total order. Identifying 
$[0,2\pi)$ with S 1 using 
$\theta\mapsto \mathrm{e}^{\theta {i}}$, and applying Remark 2.3 provides a cyclic order on S 1.
3. Cyclic orders on trees
A tree is a non-empty, connected, acyclic, simplicial graph. An edge with vertices 
$u,v$ is associated to two directed edges: (u, v) and (v, u).
In this section, we cyclically order the directed edges of a tree. We emphasize that each edge corresponds to two directed edges. The cyclic ordering arises from the following statement, which is illustrated in Figure 1.
Figure 1. A clockwise boundary path cyclically orders the directed edges.
Lemma 3.1. Let T be a finite tree embedded in the plane. There is an induced cyclic ordering on the directed edges of T.
Proof. Regarding T as a disc diagram, the clockwise boundary path 
${\partial}_{\textsf{p}}(T)$ provides an embedding of the directed edges into S 1, hence inducing a cyclic order via Remark 2.4. Note that the boundary path traverses each edge twice: once in each direction.
Definition 3.2. A tree 
$T = (V,E)$ is a c-tree if 
$\text{link}(v)$ has a cyclic order for each 
$v \in V$. Equivalently, there is a cyclic order on the edges adjacent to each vertex.
We emphasize that link(v) has a point for each edge containing v. We are not considering directed edges here.
Definition 3.3. An embedding 
$T \rightarrow {\mathbb{R}}^2$ of a locally finite c-tree is coordinated if for each 
$v \in V$ with adjacent edges 
$e_1 \prec e_2 \prec \cdots \prec e_n \prec e_1$, their images 
$\bar e_1 \prec \bar e_2 \prec \cdots \prec \bar e_n \prec \bar e_1$ are in the same clockwise order about 
$\bar v \in {\mathbb{R}}^2$.
Lemma 3.4. Each (locally) finite c-tree T has a coordinated embedding 
$T \rightarrow {\mathbb{R}}^2$.
Proof. We produce a ‘thickening’ of T into 0-handles and 1-handles to obtain a disk as follows. Embed a valence n-vertex v with cyclically ordered edges 
$e_1,\ldots, e_n$ in a unit disk, by identifying v with 0 and identifying each edge with the segment joining 0 and 
$\mathrm{e}^{\frac{2\pi}{n}{i}}$. Join disks for adjacent vertices along neighbourhoods (consistently orientated) to form a surface S homeomorphic to the unit disk, see Figure 2.
Figure 2. Handlebody decomposition of a tree in 
$\mathbb{R}^2$.
Remark 3.5. The embedding of Lemma 3.4 is unique up to ambient isotopy. Hence, for any finite subtrees 
$T_a \subset T_b$, a coordinated embedding of Ta is essentially the same as an embedding of Ta induced by a coordinated embedding of Tb. Indeed, the way Lemma 3.4 embeds Tb induces the way it embeds Ta simply by ‘forgetting’ 
$T_b - T_a$.
For any two finite subtrees, their embeddings agree with a coordinated embedding of a larger finite tree containing them.
Theorem 3.6. Let T be a c-tree. There is an induced cyclic order on the set of directed edges of T. It is uniquely determined by the cyclic orders on vertex links.
Proof. For a c-tree, take a coordinated embedding of a finite subtree Tʹ. Lemma 3.1 yields a cyclic order on the directed edges of Tʹ. This cyclic order is consistent for 
$T^{\prime}\subset T^{\prime\prime}$ whenever Tʹʹ is a larger finite subtree. Hence, it induces a cyclic order on all directed edges of T.
Uniqueness holds since the cyclic order on each 
$\operatorname{link}(v)$ is determined by the cyclic order of the outgoing directed edges at v. Note that in the cyclic ordering of the directed edges of star(v), for each edge, its two directed edges are consecutive.
Lemma 3.7. (G-invariance)
Suppose G acts on a c-tree T so that cyclic orders on vertex links are G-invariant. Then the induced cyclic order on directed edges of T is G-invariant.
Proof. This holds by Theorem 3.6 since the induced cyclic order on directed edges of T is determined by the cyclic orderings on vertex links.
4. Cyclic orders and tree augmentation
We provide an alternate explanation of the cyclic ordering on directed edges of a c-tree given in § 3. This approach constructs a correspondence between directed edges and spurs.
Definition 4.1. For vertices of a tree 
$x,y,z \in V$, the median 
$m(x,y,z)$ is the vertex equal to the intersection of geodesics 
$xy \cap yz \cap zx$.
Lemma 4.2. Let 
$T = (V, E)$ be a c-tree, there is a cyclic order on the set 
$L \subseteq V$ of spurs of T.
Proof. When 
$x,y,z \in L$ are distinct, the median 
$m = m(x,y,z)$ has three distinct edges adjacent to m pointing to 
$x, y$ and z. These edges 
$e_x, e_y$ and ez are cyclically ordered around m. Declare a cyclic order on L via:
\begin{equation*}
[x,y,z]\ \text{in}\ L \iff [e_x, e_y, e_z]\ \text{in link}(m).
\end{equation*} Non-degeneracy, cyclicity and asymmetry all follow immediately as the link of the median is cyclically ordered. For leaves 
$x,y,z,w \in L$, transitivity follows if 
$m(x,y,z) = m(x,z,w)$. Otherwise, let S be the smallest subtree containing 
$\{x,y,z,w\}$. S takes the form of an ‘H’ with two leaves at 
$m_1 = m(x,y,z)$ and two leaves at 
$m_2 = m(x,z,w)$. Via Lemma 3.4, we can embed S into the plane so that links of vertices are cyclically ordered clockwise. If 
$[x,y,z]$ and 
$[x,z,w]$ hold, then 
$[x,y,w]$ also holds, see Figure 3.
Figure 3. This explains transitivity for Lemma 4.2.
Definition 4.3. (Augmented tree) Let T be a directed c-tree, the augmented tree 
$\overline{T}$ is obtained by adding an augmented edge eaug at the barycentre of each directed edge e, see Figure 4. More precisely, for each edge 
$e \in E$, let be be its barycentre and cut e into two half edges, eout and ein. Orient the half edges so that ein and eout are incoming and outgoing at be. Under this construction links of vertices in the original tree T are unchanged, and the link of each barycentre vertex be is 
$\{e_{in}, e_{out}, e_{aug}\}$. Cyclically order 
$link(b_e)$ using the rule 
$[e_{in}, e_{out}, e_{aug}]$. Direct augmented edges away from barycentres, and note that the augmented tree 
$\overline{T}$ is now a directed c-tree.
Figure 4. The direction of e determines the position of the spur.
Theorem 4.4. There is an induced cyclic order on the set of directed edges of a c-tree T.
Proof. Construct the augmented tree 
$\overline{T}$ and note that each directed edge of T is associated to a spur of 
$\overline{T}$. Apply Lemma 4.2 to cyclically order these spurs.
5. Ordered and cyclically ordered groups
Definition 5.1. (Left-ordered group)
A group G is left-ordered if there is a strict total order 
$(G, \prec)$ such that for all 
$x, y, g \in G$ we have:
\begin{equation*}
x \prec y \ \ \implies \ \ gx \prec gy.
\end{equation*} G is left-ordered if and only if 
$G = P \sqcup \{1_G\} \sqcup N$ with 
$P P \subset P$ and 
$N N \subset N$ where 
$P = \{g \in G \ \ : \ 1_G \prec g\}$ and 
$N = \{g \in G \ \ : \ g \prec 1_G\}$. Then 
$g \prec h$ 
$\iff$ 
$g^{-1}h \in P$. The subset P is referred to as the positive cone.
Definition 5.2. (Left-cyclically ordered group)
A group G is left-cyclically ordered if there is a cyclic order on G that is left-invariant in the sense that:
\begin{equation*}[a, b, c] \ \ \implies \ \ [ga, gb, gc].\end{equation*}Remark 5.3. Let G act freely on a cyclically ordered set X. Cyclically order G via:
\begin{equation*}[a,b,c]\ \text{in}\ G \ \iff \ [ax,bx,cx]\ \text{in}\ X.\end{equation*}The following well-known statements can be found in [Reference Deroin, Navas and Rivas8, § 1.1.3] and [Reference Mann and Rivas11, Ex 2.116] respectively.
Lemma 5.4. Let G act faithfully and order-preservingly on a strict-totally ordered set 
$(X, \lt )$. Then G has an induced left-order.
Proof. Choose a well-ordering 
$\prec_w$ on X. For 
$g \neq h \in G$, let p be 
$\prec_w$-minimal with gp ≠ hp. Declare 
$g \prec h$ if gp < hp.
This relation is irreflexive as 
$gp \nless gp$. Since G acts faithfully on X, for 
$g \neq h \in G$ there exists 
$x \in X$ with gx ≠ hx, so comparability holds. G-invariance holds since 
$kgp \lt khp \ \iff \ gp \lt hp$. Let p 1 and p 2 be 
$\prec_w$-minimal with 
$x p_1 \neq y p_1$ and 
$y p_2 \neq z p_2$. If 
$p_1 = p_2$ we are done. If 
$p_1 \prec_w p_2$ then 
$y p_1 = z p_1$ and 
$x p_1 \lt z p_1$. If 
$p_2 \prec_w p_1$ then 
$x p_2 = y p_2$ and 
$x p_2 \lt z p_2$. Thus transitivity holds for 
$(G, \prec)$.
Theorem 5.5. Let G act faithfully and order-preservingly on a cyclically ordered set X. Then G has an induced left-cyclic order.
Proof. Let 
$p \in X$ and 
$\dot X$ 
$= X - \{p\}$. Observe that 
$\dot X$ is totally ordered and 
$H = stab(p)$ acts faithfully on 
$\dot X$. Via Lemma 5.4, H is left-ordered. There is a strict total order 
$(gH, \prec)$ for each left coset, by declaring 
$g \alpha \prec g \beta$ 
$\iff$ 
$\alpha \prec \beta$. This is independent of the choice g of representative, since 
$(H, \prec)$ is left H-invariant.
Our ordering on each coset provides a partial ordering on 
$G = \cup gH$. This partial ordering is G-invariant by definition. This partial ordering on G extends to a G-invariant cyclic ordering by cyclically ordering the left cosets using their bijection with Gp. Specifically 
$[a,b,c]$ holds if either:
(1)
$a \prec b \prec c$ and $ap = bp = cp$,(2)
$a \prec b$ with $ap = bp \neq cp$, or $b \prec c$ with $bp = cp \neq ap$, or $c \prec a$ with $cp = ap \neq bp$,(3)
$[ap, bp, cp]$ in X.
6. Ordering collections of cosets
Definition 6.1. (Convex subgroup)
A subgroup H of a left-ordered group 
$(G, \prec)$ is convex if for all 
$h_1,h_2\in H$ and 
$g\in G$, if 
$h_1 \prec g \prec h_2$ then 
$g \in H$.
Definition 6.2. (c-convex subgroup)
A proper subgroup H of a left-cyclically ordered group G is c-convex if there is a G-invariant cyclic order on its cosets 
$G/H$.
Two ways of defining c-convexity for subgroups of left-cyclically ordered groups appear in [Reference Clay and Ghaswala5] and [Reference Clay, Mann and Rivas7]. We refer to these as cʹ-convexity and cʹʹ-convexity and show they are both equivalent to c-convexity.
Definition 6.3. (cʹ-convex subgroup)
Let G be a left-cyclically ordered group and 
$H \subset G$ a proper subgroup. We say H is cʹ-convex if for every 
$g \notin H$ and 
$f \in G$ and 
$h_1, h_2 \in H$, if 
$[h_1, f, h_2]$ and 
$[h_1, h_2, g]$ then 
$f \in H$.
The definition of cʹʹ-convexity requires the following preliminary notion.
Definition 6.4. Let G be a left-cyclically ordered group. A proper subgroup 
$H \subset G$ is left-ordered by restriction if for each 
$h_{1},h_{2} \in H$, if 
$[h_{1}^{-1}, 1, h_{1}]$ and 
$[h_{2}^{-1}, 1, h_{2}]$ then 
$[h_{1}^{-1}h_{2}^{-1}, 1, h_{2}h_{1}]$. When 
$H \subset G$ is left-ordered by restriction, there is an induced left-order on H given by the following positive cone:
\begin{equation*}
P \ = \ \{\ h \in H \text{} \ : \ \text{} [h^{-1}, 1, h]\ \textrm{holds in}\ G \ \}.
\end{equation*}Definition 6.5. (cʹʹ-convex subgroup)
Let G be a left-cyclically ordered group. A proper subgroup 
$H \subset G$ is cʹʹ-convex if:
(1) Whenever
$h_1, h_2 \in H$ and $f \in G$, if $[h_1, 1, h_2]$ and $[h_1, f, h_2]$ then $f \in H$.(2) H is left-ordered by restriction.
Theorem 6.6. For a proper subgroup H of a left-cyclically ordered group G, the following are equivalent:
(1) c-convexity.
(2) cʹ-convexity.
(3) Property (1) of cʹʹ-convexity.
(4) cʹʹ-convexity.
Proof of (1) $\implies$ (2)
See [Reference Clay and Ghaswala5, Lemma 5.1].
Proof of (2) $\implies$ (3)
We argue by contradiction. If Property (1) fails, there exists 
$h_1, h_2 \in H$ and 
$g \notin H$ with 
$[h_2, 1, h_1]$ and 
$[h_2, g, h_1]$. Suppose 
$[1, g, h_1]$ and left-multiply by 
$h_1^{-1}$ to get 
$[h_1^{-1}, h_1^{-1}g, 1]$. Since 
$[h_1, h_1^{-1}g, 1]$ and 
$[h_1, 1, g]$, cʹ-convexity implies that 
$h_1^{-1}g \in H$, a contradiction. The case 
$[h_2, g, 1]$ is analogous.
Proof of (3) $\implies$ (1)
See [Reference Clay, Mann and Rivas7, Proposition 2.4]. We note that Property (2) of cʹʹ-convexity is not used in that proof.
Proof of (2) $\implies$ (4)
The proof that cʹ-convexity implies Property (2) of cʹʹ-convexity is shown in [Reference Clay and Ghaswala5, Lemma 5.2].
Proof of (4) $\implies$ (3)
This is immediate.
For subsets 
$U,V$ of an ordered set 
$(X, \prec)$, declare 
$U \ll V$ if there exists 
$v \in V$ with 
$u \prec v$ for all 
$u \in U$. Note that within a left-ordered group 
$(G, \prec)$ we have 
$U \ll V \iff gU \ll gV$ for all 
$g \in G$.
The following property is well known.
Lemma 6.7. Let 
$(G, \prec)$ be an ordered group and H a convex subgroup. The relation 
$\ll$ restricts to a G-invariant strict total order on the collection 
$G/H$ of left cosets.
Proof. Comparability of 
$(G/H, \ll)$ holds as cosets are disjoint and H is convex. Transitivity follows since 
$(G, \prec)$ is left-ordered. If 
$U \ll U$ for some 
$U \in G/H$, then there exists 
$v \in U$ with 
$u \prec v$ for all 
$u \in U$, so 
$v \prec v$ which is impossible.
Lemma 6.8. Let G be a left-cyclically ordered group. Let 
$H \subsetneq K \subsetneq G$ be convex subgroups. Then 
$H \ll K$ (in the induced order on K).
Proof. Let 
$(K, \prec)$ be the induced left-order of Definition 6.4. Consider a coset kH ≠ H. If 
$H \not \ll K$, there exists 
$h \in H$ with 
$k\prec h$. By Lemma 6.7, 
$k'\prec h'$ for all 
$k'\in kH$ and 
$h'\in H$. In particular, 
$k\prec 1$. Left multiplying gives 
$1\prec k^{-1}$. Since 
$H \not \ll K$, we have 
$k^{-1}\prec h^{\prime\prime}$ for some 
$h^{\prime\prime} \in H$. Finally, 
$1\prec k^{-1} \prec h^{\prime\prime}$ implies 
$k^{-1}\in H$ by convexity, a contradiction as 
$k \notin H$.
Lemma 6.9. Suppose H and K are convex subgroups of the left-cyclically ordered group G. Either 
$H \subset K$ or 
$K \subset H$.
Proof. If 
$H \not \subset K$ and 
$K \not \subset H$ then 
$H \cap K \subsetneq H$ and 
$H \cap K \subsetneq K$. Thus, 
$H \cap K \ll H$ and 
$H \cap K \ll K$ by Lemma 6.8. Thus there exists 
$h \in H$ and 
$k \in K$ with 
$\alpha \prec h$ and 
$\alpha \prec k$ for all 
$\alpha \in H \cap K$. Note that 
$h \not = k$, as otherwise 
$k \in H \cap K$ hence 
$k \prec k$. Without loss of generality, assume 
$[1, h, k]$. By convexity, 
$h \in K$. Thus, 
$h \in H \cap K$ so 
$h \prec h$, a contradiction.
Corollary 6.10. Suppose K and H are convex subgroups of a left-cyclically ordered group G. Let 
$x, y \in G$. If 
$xK \cap yH \neq \emptyset$ then either 
$xH \subset yK$ or 
$yK \subset xH$.
Proof. This follows from Lemma 6.9.
Definition 6.11. It will be convenient to consider indexed collections of subsets 
$\{H_i\}_{i \in I}$ allowing ‘repeats’ in the sense that 
$H_i = H_j$ though i ≠ j.
Although we will not use it, it seems worth articulating the following special case of our preliminary goal, Theorem 6.13.
Lemma 6.12. Let 
$(G, \prec)$ be a left-ordered group and 
$\{H_i\}_{i \in I}$ an indexed collection of convex subgroups. There is a G-invariant total order on the indexed collection of left cosets 
$\{gH_i \ : \ g \in G, i \in I\}$.
Proof. Choose a strict total order 
$\prec_{_I}$ on I. Let 
$\ll_{_I}$ denote the relation defined by:
\begin{equation*}g_1H_i \ \ll_{_I} \ g_2H_j \ \iff \
\begin{cases}
g_1H_i \neq g_2H_j\,\, \text{and } g_1H_i \ll g_2H_j
\\
g_1H_i = g_2H_j \,\, \text{and } i \prec_{_I} j.
\end{cases}\end{equation*} Transitivity and comparability of 
$\ll_{_I}$ hold since 
$(G, \prec)$ and 
$(I, \prec_{_I})$ are strict total orders. It is impossible for 
$g_1H_i \ll_{_I} g_1H_i$, as this would imply 
$i \prec_{_I} i$. Thus 
$\ll_{_I}$ is irreflexive, and therefore a strict total order.
Let 
$g_1H_i \ll_{_I} g_2H_j$. If 
$g_1H_i \neq g_2H_j$, then G-invariance of 
$(G, \prec)$ ensures 
$\alpha g_1H_i \ll_{_I} \alpha g_2H_j$ for all 
$\alpha \in G$. If 
$g_1H_i = g_2H_j$, the order depends only on 
$(I, \prec_{_I})$, and 
$\alpha g_1H_i \ll_{_I} \alpha g_2H_j$ for all 
$\alpha \in G$. Thus, 
$\ll_{_I}$ is G-invariant.
Theorem 6.13. Let G be a left-cyclically ordered group and let 
$\{H_i\}_{i \in I}$ be an indexed collection of c-convex subgroups. There is a G-invariant cyclic order on the indexed collection of left cosets 
$\{gH_i : g \in G, i \in I \}$.
Proof. Choose a strict total order 
$\prec_{_I}$ on I. For any finite subcollection of c-convex subgroups 
$\{H_j\}_{j\in J} \subseteq \{H_i\}_{i\in I}$, by Lemma 6.9 there is a chain of inclusions. (We abuse notation and regard 
$J = \{0,1,\ldots,n\}$.)
\begin{equation*}
G = H_0 \supset H_1 \supseteq \cdots \supseteq H_n.
\end{equation*} This chain of inclusions determines a graph of groups, whose underlying graph is a length-n subdivided interval. Direct all edges away from the root vertex v 0, whose vertex group is G. The edge ei terminates at the vertex vi, and 
$G_{e_i} = G_{v_i} = H_i$. As this graph of groups is telescopic its fundamental group is G.
Let 
$T = (V,E)$ be the Bass–Serre tree corresponding to this graph of groups. The vertex set 
$V = \sqcup_{i=0}^n \{gH_i \ : \ g \in G\}$ consists of the indexed collection of left cosets of vertex groups, see Figure 5.
Figure 5. Part of a finite coset tree.
There is a directed edge from 
$g_1H_k$ to 
$g_2H_{k+1}$ when 
$g_1H_k \supset g_2H_{k+1}$. Under this construction, each left coset gHj for j > 0 is represented by a directed edge.
We turn T into a directed c-tree. For the root vertex G, note that 
$\text{link}(G)$ corresponds to 
$G/H_1$ which has a G-invariant cyclic order by c-convexity. For any other vertex gHk, there is one incoming parent edge of 
$\text{link}(gH_k)$ and has outgoing edges representing containment of left-subcosets of 
$H_{k+1}$. By Lemma 6.7, 
$(H_k, \ll)$ induces a strict total order on 
$H_k / H_{k+1}$. This extends to a strict total order on 
$\{H_k\} \sqcup H_k / H_{k+1}$ by declaring Hk minimal. Translating by g provides a total order on gHk and its left 
$H_{k+1}$ cosets. This provides a cyclic order on 
$\text{link}(gH_k)$ by Remark 2.3.
Theorem 3.6 provides a cyclic order on directed edges of the c-tree T. Hence, this gives a cyclic order on left cosets of 
$\{H_j\}_{j\in J}$. This holds for any finite collection of convex subgroups. The cyclic order is consistent for graphs of groups 
$\mathcal{G}^{\prime} \subset \mathcal{G}^{\prime\prime}$ as defined above. Hence, this induces a cyclic order on all left cosets in 
$\{gH_i \ : \ g \in G, i \in I\}$. As G is the fundamental group of this graph of groups, the cyclic order on the link of each vertex is G-invariant. Hence, by Lemma 3.7 the cyclic order on left cosets is G-invariant.
Remark 6.14. The referee suggests a more self-contained proof, using the fact that G has a maximal c-convex subgroup H. The left cosets within H have a left-invariant left-order by Lemma 6.12. And this can be extended to a cyclic order on the cosets in G by combining this with cyclic order on 
$G/H$.
7. Groups acting on trees
7.1. Action on tree
Definition 7.1. An inclusion 
$H \rightarrow K$ of a left-ordered group into a left-cyclically ordered group is order-preserving if
\begin{equation*} a\prec b \prec c \ \textrm{in } H \ \ \implies \ \ [a,b,c] \ \textrm{in } K. \end{equation*}Theorem 7.2. Let G act without inversions on a tree 
$T=(V,E)$. Suppose:
(1) The stabilizer Gv is left-cyclically ordered for each vertex
$v\in V$.(2) The stabilizer Ge is left-ordered for each edge
$e\in E$.(3) The inclusion
$G_e\subset G_v$ is c-convex whenever v is a vertex of e.
Then there is a c-tree 
$\widetilde T=(\widetilde V,\widetilde E)$ such that:
(1) There exists a spur
$\tilde{e} \in \widetilde{E}$ such that $G \tilde{e}$ is a free orbit.(2) There is a G-invariant cyclic order on the orbit
$G\tilde{e}$ that induces a cyclic order on G.(3) For each
$e\in E$, the order on Ge is induced by the action of Ge on $\widetilde T$.(4) For each
$v\in V$, the cyclic order on Gv is induced by the action of Gv on $\widetilde T$.
Proof. Build 
$\widetilde{T}$ from T as follows. For each 
$v \in V$ add a spur to v for each element of the stabilizer Gv. These spurs are in correspondence with cosets of the trivial subgroup of Gv which is a c-convex subgroup. 
$G_e \subset G_v$ is c-convex by hypothesis. For each 
$v \in V$, cyclically order 
$\text{link}(v)$ via Theorem 6.13. Thus 
$\widetilde{T}$ is a c-tree.
Let 
$\tilde{e}$ be an added spur. Cyclically order the spurs and hence 
$G \tilde{e}$ by Theorem 4.4. By Lemma 3.7, the cyclic order on 
$G \tilde{e}$ is G-invariant. Finally, since G acts freely on 
$G \tilde{e}$, Remark 5.3 provides a left-cyclic order on G.
7.2. Graph of groups statement
Corollary 7.3. Let G split as a graph Γ of groups. Suppose each vertex group Gv is left-cyclically ordered, and each edge group Ge is left-ordered. Suppose each inclusion 
$G_e \hookrightarrow G_v$ of an edge group is c-convex. Then G has a left-cyclic order that restricts to the cyclic order of each vertex group Gv.
Proof. Let 
$T=(V,E)$ be the Bass–Serre tree over Γ, which we assume to be directed. V consists of all left cosets of vertex groups of Γ in G, and E consists of left cosets of edge groups of Γ in G. That is, allowing for repeats (of edge or vertex groups):
\begin{align*}
V &= \{gG_v \ : \ g \in G, \ v \in \text{Vertices}(\Gamma)\}\\
E &= \{gG_e \ : \ g \in G, \ e \in \text{Edges}(\Gamma)\}.
\end{align*} Varying 
$g \in G$, there is an edge gGe directed from gGu to gGv in T precisely when e is directed from u to v in Γ.
The stabilizer of a vertex gGv equals 
$gG_vg^{-1}$, and similarly the stabilizer of an edge gGe equals 
$gG_eg^{-1}$. Conjugation preserves the cyclic orders on Gv for each vertex, and similarly preserves the orderings on Ge for each edge, thus vertex and edge stabilizers are cyclically ordered. Let 
$\widetilde{T}$ be the c-tree obtained from T by Theorem 7.2 and note that the cyclic order on each vertex group is induced by its 
$\widetilde{T}$ action. 
$\widetilde{T}$ has a spur 
$\tilde{e}$ with a free G-orbit which provides a cyclic order on G by Remark 5.3.
We note that [Reference Chiswell6] contains an analogous result to Corollary 7.3 for a graph of groups with left-orderable vertex groups and convex edge groups.
Remark 7.4. Every group acting faithfully without inversions on a c-tree arises as in Corollary 7.3. The edge stabilizers are c-convex subgroups of the vertex stabilizers. Indeed, for each edge e at a vertex v, the left cosets of stab(e) in Gv correspond Gv-equivariantly to the edges in the Gv-orbit of e. The Gv-invariant cyclic order on the edges yields a Gv-invariant cyclic ordering on the cosets. Finally, every action on a tree arises as the Bass–Serre tree of a graph of groups.
Acknowledgements
We are extremely grateful to the referee for greatly improving the exposition, correcting our mistakes and connecting us to the literature.
Competing interests
The authors declare none.
