Strong digraph groups

Abstract

A digraph group is a group defined by non-empty presentation with the property that each relator is of the form $R(x, y)$, where x and y are distinct generators and $R(\cdot , \cdot )$ is determined by some fixed cyclically reduced word $R(a, b)$ that involves both a and b. Associated with each such presentation is a digraph whose vertices correspond to the generators and whose arcs correspond to the relators. In this article, we consider digraph groups for strong digraphs that are digon-free and triangle-free. We classify when the digraph group is finite and show that in these cases it is cyclic, giving its order. We apply this result to the Cayley digraph of the generalized quaternion group, to circulant digraphs, and to Cartesian and direct products of strong digraphs.

1 Introduction

Given a finite digraph $\Gamma $ with vertex set $V(\Gamma )$ and arc set $A(\Gamma )$ (without loops or parallel arcs) and an element $R(a,b)$ in the free group of rank 2 generated by a and b, the digraph group $G_{\Gamma} (R)$ is the group defined by the presentation

$$\begin{align*}P_{\Gamma}(R) = \langle {x_v\ (v\in V(\Gamma))}\ |\ {R(x_u,x_v)\ ([u,v]\in A(\Gamma))} \rangle.\end{align*}$$

These groups were introduced in [7] and contain the graph groups or right angled Artin groups as special cases (by setting $R(a,b)=aba^{-1}b^{-1}$ ). Formal definitions of undefined terms of the introduction are given in Section 2.

Digraph groups with balanced presentations (i.e., digraph groups corresponding to digraphs with an equal number of vertices and arcs) were considered in [7] and digraph groups corresponding to digraphs with one more arc than vertices were considered in [5]. Two families of finite, noncyclic, digraph groups were obtained in [6, Corollaries A2 and B2]. In this article, we consider digraph groups $G_{\Gamma} (R)$ where $\Gamma $ is a strong digraph that is digon-free and triangle-free. We state our results in terms of the period of the digraph (that is, the greatest common divisor of the lengths of its cycles), the one-relator group $K=\langle {a,b}\ |\ {R(a,b)} \rangle $ , and integers $\alpha ,\beta $ , which denote the exponent sums of a and of $b^{-1}$ in $R(a,b)$ , respectively.

In this context, [17, Theorem 3] concerns digraph groups $G_{\Gamma} (R)$ where $\Gamma $ is a cycle of length at least 4, and can be expressed as follows:

Theorem 1.1 [17]

Let $\Gamma $ be a cycle of length $n\geq 4$ and let $R(a, b)$ be a cyclically reduced word that involves both a and b with exponent sums $\alpha $ and $\beta $ in a and $b^{-1}$ , respectively, and let $K=\langle {a,b}\ |\ {R(a,b)} \rangle $ . Then $G_{\Gamma} (R)$ is finite if and only if $\alpha \neq 0$ , $\beta \neq 0$ , ${\mathrm {gcd}\{\alpha ,\beta \}=1}$ , $\alpha ^n\neq \beta ^n$ , $a^\alpha =b^\beta $ in K, in which case $G_{\Gamma} (R)$ is cyclic of order $|\alpha ^n-\beta ^n|$ .

When $\Gamma $ is a cycle, the digraph group $G_{\Gamma} (R)$ is an example of a cyclically presented group [12, Chapter III, Section 9]. As observed in [17], Theorem 1.1 cannot be directly extended to the cases $n=2$ and $n=3$ , that is, to the cases where $\Gamma $ is neither digon-free nor triangle-free. Examples that demonstrate this include the Macdonald groups $Mac(a,a)$ which, for $a\in \mathbb {Z}$ , $a\neq 0,1,2$ , are finite of rank 2 [13, 16]; the Fox groups $\langle {x,y}\ |\ {xy^n=y^lx, yx^n=x^ly} \rangle $ which are finite of rank 2 if $(n,l)=1$ , $n\neq l$ [4]; the Mennicke groups $M(q,q,q)$ , which are finite of rank 3 for each $q\geq 3$ [15]; and the Johnson groups $J(q,q,q)$ which are finite of rank 3 for each even $q\geq 2$ [11], [12, p. 70]. A strong digraph is one in which there is a path joining every pair of vertices, and so the cycle of length n is a strong digraph of period n. In this article, we generalize Theorem 1.1 by replacing the cycle of length at least 4 by a nontrivial, strong digraph that is digon-free and triangle-free. Our main result is the following:

Theorem 1.2 Let $\Gamma $ be a nontrivial, strong digraph of period p that is digon-free and triangle-free and let $R(a,b)$ be a cyclically reduced word that involves both a and b and which has exponent sums $\alpha ,\beta $ in a and $b^{-1}$ , respectively, and let $K=\langle {a,b}\ |\ {R(a,b)} \rangle $ . Then $G_{\Gamma} (R)$ is finite if and only if $\alpha \neq 0$ , $\beta \neq 0$ , $\mathrm {gcd}\{\alpha ,\beta \}=1$ , $\alpha ^p\neq \beta ^p$ , $a^\alpha =b^\beta $ in K, in which case $G_{\Gamma} (R)$ is cyclic of order $|\alpha ^p-\beta ^p|$ .

In Section 5, we apply this to the Cayley digraph of the generalized quaternion group, to circulant digraphs, and to Cartesian and direct products of strong digraphs.

2 Preliminaries

A digraph (or directed graph) $\Gamma $ consists of a finite set $V(\Gamma )$ of vertices and a set $A(\Gamma )$ of arcs, which are ordered pairs $[u,v]$ of distinct vertices (as such, $\Gamma $ does not contain parallel arcs or loops); it is nontrivial if it has at least two vertices. The underlying graph of $\Gamma $ is the graph with vertex set $V(\Gamma )$ and edge set $E(\Gamma )$ consisting of all unordered pairs $\{u,v\}$ (edges), where $[u,v]\in A(\Gamma )$ . A digon is a subdigraph of $\Gamma $ consisting of vertices $u,v\in V(\Gamma )$ and arcs $[u,v],[v,u] \in A(\Gamma )$ . A triangle is a subdigraph of $\Gamma $ consisting of vertices $u,v,w\in V(\Gamma )$ and either arcs $[u,v],[v,w],[w,u]\in A(\Gamma )$ or arcs $[u,v],[v,w],[u,w]\in A(\Gamma )$ . A digraph is said to be digon-free (resp. triangle-free) if it contains no digons (resp. triangles). A tournament is a digraph $\Gamma $ in which exactly one of $[u,v],[v,u]\in A(\Gamma )$ for each pair of vertices $u,v\in V(\Gamma )$ . A walk (of length $n-1$ ) in a digraph $\Gamma $ is a collection of vertices $v_1,v_2,\dots ,v_n$ such that $[v_i,v_{i+1}]\in A(\Gamma )$ for each $1\leq i<n$ ; it is closed if $v_n=v_1$ . We denote such a walk $v_1\rightarrow v_2 \rightarrow \cdots \rightarrow v_n$ . A path is a walk in which the vertices are distinct. A cycle (of length n) is a collection of distinct vertices $v_1,v_2,\dots ,v_n$ such that $[v_i,v_{i+1}]\in A(\Gamma )$ for each $1\leq i<n$ and $[v_n,v_1]\in A(\Gamma )$ . The period $p(\Gamma )$ of a digraph $\Gamma $ is the greatest common divisor of the lengths of its cycles, and so $p(\Gamma )$ is equal to the greatest common divisor of the lengths of its closed walks. A digraph $\Gamma $ is strong (or strongly connected) if for each $u,v\in V(\Gamma )$ there is a path from u to v (and hence, also a path from v to u); it follows that every vertex of a nontrivial, strong digraph is contained in some cycle [9, p. 64]. A weak component (or weakly connected component) of a digraph $\Gamma $ is a maximal subdigraph of $\Gamma $ whose underlying graph is connected. A digraph is weak (or weakly connected) if it has exactly one weak component. A digraph $\Gamma $ is bipartite if there is a vertex partition such that if $[u,v]\in A(\Gamma )$ then either $u\in V_1$ and $v \in V_2$ or $u\in V_2$ and $v \in V_1$ . A strong digraph is bipartite if and only if it has no odd length cycles [9, Theorem 6.14], that is, if and only if its period is even.

The Cartesian product $\Gamma _1\square \Gamma _2$ of digraphs $\Gamma _1,\Gamma _2$ is the digraph $\Gamma $ with ${V(\Gamma )=V(\Gamma _1)\times V(\Gamma _2)}$ and $A(\Gamma ) = \{ [ (u,u'), (v,v') ]\ |\ [u,v]\in A(\Gamma _1), u'=v'~\mathrm {or} [u',v']\in A(\Gamma _2), u=v \}$ . The direct product $\Gamma _1\times \Gamma _2$ of digraphs $\Gamma _1,\Gamma _2$ is the digraph $\Gamma $ with $V(\Gamma )=V(\Gamma _1)\times V(\Gamma _2)$ and $A(\Gamma ) = \{ [ (u,u'), (v,v') ]\ |\ [u,v]\in A(\Gamma _1)~\mathrm {and}~[u',v']\in A(\Gamma _2)\}$ . Both graph products $\square $ and $\times $ are associative [8].

Given a finite group G with generating set S that does not contain the identity of G, the Cayley digraph $\mathrm {Cay}(G,S)$ is the digraph $\Gamma $ with $V(\Gamma )=G$ and arc set

$$\begin{align*}A(\Gamma) = \{ [g,h]\ |\ h=gs~\mathrm{for~some}~s\in S\}.\end{align*}$$

Thus, every finite Cayley digraph is strong. The circulant digraph $\mathrm {circ}_n\{ d_1,\dots ,d_t\}$ , where $n\geq 2$ and $1\leq d_1,\dots ,d_t<n$ are distinct integers, is the digraph $\Gamma $ with ${V(\Gamma )=\{0,1,\dots ,n-1\}}$ and arcs $[i,i+d_j]$ for each $0\leq i<n, 1\leq j\leq t$ (where the entries are taken $\bmod \,n$ ) [1]. It is weakly connected if and only if ${\mathrm {gcd}\{n,d_1,\dots ,d_t\}=1}$ , in which case it is strongly connected and is the Cayley digraph $\mathrm {Cay}(\mathbb {Z}_n, \{d_1,\dots ,d_t\})$ .

3 Digraph groups

First, we observe that if $\Gamma $ has weakly connected components $\Gamma _1, \dots , \Gamma _k$ with ${k\geq 2}$ , then the presentation $P_{\Gamma} (R)$ decomposes as the disjoint union of the presentations $P_{\Gamma _1}(R),\dots , P_{\Gamma _k}(R)$ , so $G_{\Gamma} (R)$ is isomorphic to the free product $G_{\Gamma _1}(R)*\cdots * G_{\Gamma _k}(R)$ . Therefore, without loss of generality, in considering digraph groups $G_{\Gamma} (R)$ , we may assume that $\Gamma $ is weak.

Lemma 3.1 [17]

Let $\Gamma $ be a nontrivial, weak digraph that is digon-free and triangle-free and let $R(a,b)$ be a cyclically reduced word that involves both a and b and which has exponent sums $\alpha ,\beta $ in a and $b^{-1}$ , respectively, and let $K=\langle {a,b}\ |\ {R(a,b)} \rangle $ . If $G_{\Gamma} (R)$ is finite, then $\alpha \neq 0, \beta \neq 0$ , and $a^\alpha =b^\beta $ in K.

Proof By [17, Theorem 4], if K has Property $W_1$ (see [17] for the definition), then $G_{\Gamma} (R)$ is infinite, so we may assume that K does not satisfy Property $W_1$ . Then, by [17, Proposition, p. 248], we have $\alpha \neq 0$ , $\beta \neq 0$ , and $a^\alpha =b^\beta $ in K (see [7, p. 5] for further discussion on this point).

Lemma 3.2 Let $\Gamma $ be a digraph and let $R(a,b)$ be a cyclically reduced word that involves both a and b and which has exponent sums $\alpha ,\beta $ in a and $b^{-1}$ , respectively, and let $K=\langle {a,b}\ |\ {R(a,b)} \rangle $ . If $a^\alpha =b^\beta $ in K, then $G_{\Gamma} (R)$ is a quotient of $G_{\Gamma} (a^\alpha b^{-\beta })$ , so, in particular, if $G_{\Gamma} (a^\alpha b^{-\beta })$ is abelian, then $G_{\Gamma} (R) \cong G_{\Gamma} (a^\alpha b^{-\beta })$ .

Proof Since $a^\alpha =b^{\beta }$ in K, the presence of a relator $R(x_u,x_v)$ in $P_{\Gamma} (R)$ implies that the relation $x_u^\alpha = x_v^{\beta }$ holds in $G_{\Gamma} (R)$ . Therefore, the corresponding relators $x_u^\alpha x_v^{-\beta }$ can be added to the defining presentation for $G_{\Gamma} (R)$ , so $G_{\Gamma} (R)$ is a quotient of $G_{\Gamma} (a^\alpha b^{-\beta })$ . If $G_{\Gamma} (a^\alpha b^{-\beta })$ is abelian, then $G_{\Gamma} (R)$ is abelian and so the relators $R(x_u,x_v)$ are equivalent to the relators $x_u^{\alpha }x_v^{-\beta }$ , so can be removed, that is, ${G_{\Gamma} (R)\cong G_{\Gamma} (a^\alpha b^{-\beta })}$ .

Lemma 3.3 [17]

Let $\Gamma $ be a digraph that is not a tournament, and let $R(a,b)$ be a cyclically reduced word which has exponent sums $\alpha ,\beta $ in a and $b^{-1}$ , respectively. If $G_{\Gamma} (R)$ is finite, then $\mathrm {gcd}\{\alpha ,\beta \}=1$ .

Proof Since $\Gamma $ is not a tournament, there exists a pair of vertices $u,v\in V(\Gamma )$ that are not connected by an arc. As in [17, p. 248] (or [7, Proof of Lemma 3.3]), killing all generators of $G_{\Gamma} (R)$ except $x_u,x_v$ and then adjoining relators $x_u^{d},x_v^{d}$ , where ${d=\mathrm {gcd}\{\alpha ,\beta \}}$ , gives that $G_{\Gamma} (R)$ maps onto the group $\langle {x_u,x_v}\ |\ {x_u^{d},x_v^{d}} \rangle \cong \mathbb {Z}_{d}*\mathbb {Z}_{d}$ , which is infinite if $d>1$ .

Lemma 3.4 Let $\Gamma $ be a digraph, and let $R(a,b)$ be a cyclically reduced word which has exponent sums $\alpha ,\beta $ in a and $b^{-1}$ , respectively. Then $G_{\Gamma} (R)$ maps epimorphically to $\mathbb {Z}_{|\alpha -\beta |}$ . In particular, if $\alpha =\beta $ , then $G_{\Gamma} (R)$ is infinite.

Proof Let $\phi :G_{\Gamma} (R)\rightarrow \mathbb {Z}_{|\alpha -\beta |}$ be given by $\phi (x_v)=1\in \mathbb {Z}_{|\alpha -\beta |}$ for each $v\in V(\Gamma )$ . Then, for each relator $R(x_u,x_v)$ of $G_{\Gamma} (R)$ , we have $\phi (R(x_u,x_v))=\alpha \cdot 1 -\beta \cdot 1 =0 \in \mathbb {Z}_{|\alpha -\beta |}$ , so $\phi $ is a homomorphism, and since $\phi (x_v)=1$ for some $v\in V(\Gamma )$ , it is an epimorphism.

Lemma 3.5 Let $\Gamma $ be a bipartite digraph and let $R(a,b)$ be a cyclically reduced word which has exponent sums $\alpha ,\beta $ in a and $b^{-1}$ , respectively, and suppose $\mathrm {gcd}\{\alpha ,\beta \}=1$ . Then $G_{\Gamma} (R)$ maps epimorphically to $\mathbb {Z}_{|\alpha ^2-\beta ^2|}$ . In particular, if $\alpha =\pm \beta $ , then $G_{\Gamma} (R)$ is infinite.

Proof Suppose $\Gamma $ has vertex partition . Let $\phi :G_{\Gamma} (R) \rightarrow \mathbb {Z}_{|\alpha ^2-\beta ^2|}$ be given by $\phi (x_u)=\alpha $ if $u\in V_1$ and $\phi (x_u)=\beta $ if $u \in V_2$ . Then given an arc $[u,v]$ of $G_{\Gamma} (R)$ , we have $\phi (R(x_u,x_v))=\alpha ^2-\beta ^2=0$ if $u\in V_1$ and ${\phi (R(x_u,x_v))=\alpha \beta -\beta \alpha =0}$ if $u\in V_2$ . Thus, $\phi $ is a homomorphism. Since $\mathrm {gcd}\{\alpha ,\beta \}=1$ , there exist $r,s\in \mathbb {Z}$ such that $r\alpha +s\beta =1$ , and so if $u \in V_1, v \in V_2$ , then $\phi (x_u^rx_v^s)=r\alpha +s\beta =1\in \mathbb {Z}_{|\alpha ^2-\beta ^2|}$ , so $\phi $ is an epimorphism.

Lemma 3.6 Let $\Gamma $ be a nontrivial, weak digraph that is digon-free and triangle-free, and let $R(a,b)$ be a cyclically reduced word that involves both a and b and which has exponent sums $\alpha ,\beta $ in a and $b^{-1}$ , respectively, and suppose $\alpha =-\beta $ . Then ${G=G_{\Gamma} (R)}$ is finite if and only if $\alpha =-\beta = \pm 1$ , $a^\alpha =b^\beta $ in K, and $\Gamma $ is not bipartite, in which case $G\cong \mathbb {Z}_2$ .

Proof Suppose G is finite. Then Lemma 3.1 implies $\alpha \neq 0, \beta \neq 0$ , and $a^\alpha =b^\beta $ in K, and Lemma 3.3 implies $\mathrm {gcd}\{\alpha ,\beta \}=1$ , so $\alpha =-\beta =\pm 1$ . Moreover, $\Gamma $ is not bipartite by Lemma 3.5. Under these conditions $a=b^{-1}$ in K and so, given an arc $[u,v]\in A(\Gamma )$ , we have $x_u=x_v^{-1}$ in G. Fix a vertex $w\in V(\Gamma )$ . Then, since the underlying graph of $\Gamma $ is connected, for any $v\in V(\Gamma )$ we have either $x_v=x_w$ or $x_v=x_w^{-1}$ . Therefore, G is cyclic. Since $\Gamma $ is not bipartite, there is an odd length cycle $1\rightarrow 2\rightarrow 3\rightarrow \dots \rightarrow r\rightarrow 1$ , say. Then $x_1=x_2^{-1}=x_3=\dots =x_{r-1}^{-1}=x_r=x_1^{-1}$ , so $x_1^2=1$ . Moreover, each generator of G is equal to any other generator or its inverse, so G is generated by $x_1$ , which satisfies the relation $x_1^2=1$ , that is, G is cyclic of order at most 2. Then, by Lemma 3.4, G maps onto $\mathbb {Z}_2$ , so $G\cong \mathbb {Z}_2$ .

By Lemma 3.6, we may assume $\alpha \neq -\beta $ . In this situation, we summarize Lemmas 3.1–3.4 in the following result.

Theorem 3.7 Let $\Gamma $ be a nontrivial, weak digraph that is digon-free and triangle-free and let $R(a,b)$ be a cyclically reduced word that involves both a and b and which has exponent sums $\alpha ,\beta $ in a and $b^{-1}$ , respectively, where $\alpha \neq -\beta $ , and let $K=\langle {a,b}\ |\ {R(a,b)} \rangle $ . If $G_{\Gamma} (R)$ is finite, then $\alpha \neq 0, \beta \neq 0$ , $\alpha \neq \beta $ , $\mathrm {gcd}\{\alpha ,\beta \}=1$ , and $a^\alpha =b^\beta $ in K, in which case $G_{\Gamma} (R)$ is a quotient of $G_{\Gamma} (a^\alpha b^{-\beta })$ , so, in particular, if $G_{\Gamma} (a^\alpha b^{-\beta })$ is abelian, then $G_{\Gamma} (R) \cong G_{\Gamma} (a^\alpha b^{-\beta })$ .

4 Strong digraph groups

In this section, we prove Theorem 1.2.

Lemma 4.1 Let $\Gamma $ be a digraph, and let $G=G_{\Gamma} (a^{\alpha } b ^{-\beta })$ where $\mathrm {gcd}\{\alpha ,\beta \}=1$ . If v is a vertex of a cycle of length $\Delta $ in $\Gamma $ , then $x_v^{|\alpha ^\Delta - \beta ^\Delta |}=1$ in G.

Proof This is essentially stated implicitly in [17, p. 248], and proofs are given in [7, Lemma 3.4] and [3, Lemma 3.4].

Lemma 4.2 Let $\Gamma $ be a digraph, and let $G=G_{\Gamma} (a^{\alpha } b^{-\beta })$ where $\mathrm {gcd}\{\alpha ,\beta \}=1$ . Let $u\in V(\Gamma )$ be a vertex of some cycle, and suppose there is a path from u to a vertex w. Then the generator $x_u$ of G is equal to a power of the generator $x_w$ .

Proof Label the path from u to w as $u=1 \rightarrow 2 \rightarrow 3 \rightarrow \cdots \rightarrow (n-1) \rightarrow n=w$ . First, consider the arc $[1,2]$ . Since $1$ is the vertex of some cycle, of length $\Delta $ , say, Lemma 4.1 implies $x_1^\gamma =1$ , where $\gamma =|\alpha ^\Delta - \beta ^\Delta |$ . Now $\mathrm {gcd}\{\alpha ,\gamma \}=1$ , so there exist $r,s\in \mathbb {Z}$ such that $r\alpha +s \gamma =1$ , so $r\alpha \equiv 1 \bmod \gamma $ . Moreover, there is a relation $x_1^\alpha =x_2^\beta $ in $\Gamma $ , so

$$\begin{align*}x_1=x_1^{r\alpha+s \gamma}= (x_1^\alpha)^r(x_1^\gamma)^s=(x_2^\beta)^r=x_2^{r\beta}.\end{align*}$$

Therefore, $x_1$ is equal to a power of $x_2$ . Moreover, $x_1^\gamma =1$ , so $(x_2^{r\beta })^\gamma =1$ , i.e., $x_2^{r\beta \gamma }=1$ . But also $x_1^\alpha =x_2^\beta $ so $(x_2^{r\beta })^\alpha =x_2^\beta $ , so $x_2^{(1-r\alpha )\beta }=1$ , i.e., $x_2^{s\beta \gamma }=1$ . Thus, $x_2^{(r\beta \gamma ,s\beta \gamma )}=1$ , i.e., $x_2^{|\beta \gamma |}=1$ . (The argument in this paragraph is that of [7, Proof of Lemma 3.1].)

Now consider the arc $[2,3]$ . Let $\gamma '=\beta \gamma $ . Noting that $\mathrm {gcd}\{\alpha ,\gamma '\}=1$ , repeating the argument of the previous paragraph gives that $x_2$ is equal to a power of $x_3$ and ${x_3^{|\beta \gamma '|}=1}$ , that is, $x_3^{|\beta ^2 \gamma |}=1$ . Continuing in this way, we see that for each $1\leq i<n$ the generator $x_i$ is equal to a power of $x_{i+1}$ and $x_i^{|\beta ^{i-1}\gamma |}=1$ . Therefore, $x_1$ is equal to a power of $x_n$ , that is, $x_u$ is equal to a power of $x_w$ , as required.

Lemma 4.3 Let $\Gamma $ be a nontrivial, strong digraph of period p, and let $\alpha ,\beta \in \mathbb {Z}$ satisfy $\alpha \neq 0$ , $\beta \neq 0$ , $\alpha \neq \pm \beta $ , $\mathrm {gcd}\{\alpha ,\beta \}=1$ . Then $G_{\Gamma} (a^\alpha b^{-\beta })$ is finite and cyclic, of order dividing $|\alpha ^p-\beta ^p|$ .

Proof Fix a vertex $w\in V(\Gamma )$ . Since $\Gamma $ is nontrivial and strong, for every vertex $u\in V(\Gamma )$ , u is a vertex of some cycle and there is a walk from u to w. Therefore, by Lemma 4.2, the generator $x_u$ is equal to some power of $x_w$ . Therefore, every generator of G is equal to some power of $x_w$ , so G is cyclic, generated by $x_w$ . Since the choice of w was arbitrary, G is cyclic and generated by $x_v$ for any $v \in V(\Gamma )$ and, by Lemma 4.1, if $\Delta $ is the length of any cycle of which v is a vertex, $x_v^{|\alpha ^\Delta -\beta ^\Delta |}=1$ in G. That is, G is cyclic of order dividing $f(\Delta )=|\alpha ^\Delta -\beta ^\Delta |$ . Applying this observation to every vertex $v\in V(\Gamma )$ and every cycle of $\Gamma $ of which v is a vertex, we see that G is cyclic of order dividing $\mathrm {gcd} \{ f(\Delta )\ |\ \Delta ~\mathrm {is~the~length~of~some~cycle~of}~\Gamma \}$ . That is, G is cyclic of order dividing $|\alpha ^p-\beta ^p|$ .

Lemma 4.4 Let $\Gamma $ be a strong digraph of period $p\geq 2$ , let $\Lambda $ be the cycle of length p, and let $R(a,b)$ be a cyclically reduced word. Then $G_{\Gamma} (R)$ maps epimorphically to $G_{\Lambda} (R)$ .

Proof (In this proof, subscripts of $v_*,V_*$ , and $y_*$ terms are to be taken $\bmod p$ .) By [2, Theorem 10.5.1], there exists a vertex partition such that if $[u,v]\in A(\Gamma )$ then $u \in V_i$ and $v \in V_{i+1}$ for some $0\leq i<p$ . By adjoining all relators $R(x_{v_i},x_{v_{i+1}})$ , where $v_i\in V_i, v_{i+1}\in V_{i+1}$ to the defining presentation for $G=G_{\Gamma} (R)$ , we see that G maps onto

$$\begin{align*}H=\langle { x_{v_i}\ (v_i \in V_i, 0\leq i<p)}\ |\ {R(x_{v_i}, x_{v_{i+1}})\ (v_i \in V_i, v_{i+1} \in V_{i+1}, 0\leq i<p)} \rangle.\end{align*}$$

For each $0\leq i<p$ , equating all generators $x_{v_i}$ ( $v_i\in V_i$ ), and introducing generators $y_i=x_{v_i}$ , we see that H maps onto

$$\begin{align*} K&= \Bigg \langle \begin{array}{l} {x_{v_i}\ (v_i \in V_i, 0\leq i<p),}\\{y_i\ (0\leq i<p)} \end{array} \Bigg | \begin{array}{l} {R(x_{v_i}, x_{v_{i+1}})\ (v_i \in V_i, v_{i+1} \in V_{i+1}, 0\leq i<p),}\\{y_i=x_{v_i} \ (v_i \in V_i, 0\leq i<p)} \end{array} \Bigg \rangle\\ &= \langle {y_i\ (0\leq i<p)}\ |\ {R(y_i,y_{i+1})\ (0\leq i<p)} \rangle\\ &= G_{\Lambda} (R), \end{align*}$$

where $V(\Lambda )=\{ 0,1,\dots ,p-1 \}$ and $A(\Lambda )=\{[i,i+1]\ |\ 0\leq i<p\}$ .

We can now prove Theorem 1.2.

Proof of Theorem 1.2

Suppose first $\alpha =-\beta $ . Then, by Lemma 3.6, $G=G_{\Gamma} (R)$ is finite if and only if $\alpha =\pm 1$ , $\Gamma $ is not bipartite, and $a^\alpha =b^\beta $ in K, in which case $G\cong \mathbb {Z}_2$ . Equivalently, G is finite if and only if $\alpha \neq 0, \beta \neq 0$ , $\mathrm {gcd}\{\alpha ,\beta \}=1$ , $\alpha ^p\neq \beta ^p$ , and $a^\alpha =b^\beta $ in K, in which case $G\cong \mathbb {Z}_{|\alpha ^p-\beta ^p|}$ .

Suppose then $\alpha \neq -\beta $ . If G is finite, then Theorem 3.7 implies $\alpha \neq 0,\beta \neq 0, \alpha \neq \beta $ (equivalently, $\alpha ^p \neq \beta ^p$ ), $\mathrm {gcd}\{\alpha ,\beta \}=1$ , and $a^\alpha =b^\beta $ in K. Under these conditions, Lemma 4.3 implies that $G_{\Gamma} (a^\alpha b^{-\beta })$ is finite and cyclic of order dividing $\gamma =|\alpha ^p-\beta ^p|$ . Then $G\cong G_{\Gamma} (a^\alpha b^{-\beta })$ by Theorem 3.7. We now show that G maps epimorphically to $\mathbb {Z}_\gamma $ . If $p=1$ , this follows from Lemma 3.4, so assume $p\geq 2$ . By Lemma 4.4, $G=G_{\Gamma} (a^\alpha b^{-\beta })$ maps epimorphically to $G_{\Lambda} (a^\alpha b^{-\beta })$ , where $\Lambda $ is the cycle of length p. But $G_{\Lambda} (a^\alpha b^{-\beta })\cong \mathbb {Z}_\gamma $ by [17, p. 248] (or [7, Lemma 3.4], [3, Lemma 3.4]). Thus, $G\cong \mathbb {Z}_\gamma $ , as required.

5 Applications

In this section, we obtain corollaries to Theorem 1.2 for particular types of strong digraphs.

5.1 A Cayley digraph

We consider the Cayley digraph of the generalized quaternion group

$$\begin{align*}Q_{2n}=\langle {c,d}\ |\ {c^{2n}, c^nd^{-2}, cdcd^{-1}} \rangle\end{align*}$$

with respect to the generating set $\{c,d\}$ .

Lemma 5.1 Let $\Gamma = \mathrm {Cay}(Q_{2n},\{c,d\})$ where $n\geq 2$ . Then $\Gamma $ is digon-free and triangle-free and the period $p(\Gamma )=\mathrm {gcd}\{n,2\}$ .

Proof Since $c^2, d^2, c^3, d^3, c^2d^{\pm 1}, d^2c^{\pm 1}\neq e$ in $Q_{2n}$ (where e denotes the identity) the digraph $\Gamma $ is digon-free and triangle-free. It contains the following cycles:

• • $e\rightarrow d\rightarrow {d^2}\rightarrow {d^3}\rightarrow {d^4}={c^{2n}}=e$ ,

• • $e \rightarrow c \rightarrow {c^2} \rightarrow \cdots \rightarrow {c^n} \rightarrow {c^nd} \rightarrow {c^nd^2}= {c^nc^n}={c^{2n}}=e$ ,

• • ${e}\rightarrow {c}\rightarrow {c^2}\rightarrow \cdots \rightarrow {c^{2n-1}}\rightarrow {c^{2n-1}d}\rightarrow {c^{2n-1}dc} = {c^{2n-2}d}\rightarrow {c^{2n-2}dc} = {c^{2n-3} d}\rightarrow \cdots \rightarrow {c^nd}\rightarrow {c^nd^2} ={c^nc^n} = {c^{2n}} ={e}$ ,

of lengths $4,n+2,3n$ , respectively, and so $p=p(\Gamma )$ divides $\mathrm {gcd}\{4,n+2,3n\}$ , so $p=1$ if n is odd and $p|2$ if n is even. If n is even, then $\Gamma $ is bipartite with vertex partition

and so $\Gamma $ has no odd length cycles and hence $p=2$ .

Noting that Cayley digraphs are strong, we obtain the following.

Corollary 5.2 Let $\Gamma = \mathrm {Cay}(Q_{2n},\{c,d\})$ where $n\geq 2$ , let $p=\mathrm {gcd}\{n,2\}$ , and let $R(a,b)$ be a cyclically reduced word that involves both a and b and which has exponent sums $\alpha ,\beta $ in a and $b^{-1}$ , respectively, and let $K=\langle {a,b}\ |\ {R(a,b)} \rangle $ . Then $G=G_{\Gamma} (R)$ is finite if and only if $\alpha \neq 0, \beta \neq 0$ , $\mathrm {gcd}\{\alpha ,\beta \}=1$ , $\alpha ^p\neq \beta ^p$ , $a^\alpha =b^\beta $ in K, in which case G is cyclic of order $|\alpha ^p-\beta ^p|$ .

5.2 Circulant digraphs

Lemma 5.3 Let $\Gamma = \mathrm {circ}_n\{d_1,\dots ,d_t\}$ , where $n\geq 2$ and $1\leq d_i<n$ for each $1\leq i\leq t$ , and where $\mathrm {gcd}\{n,d_1,\dots ,d_t\}=1$ . Then $p(\Gamma )=\mathrm {gcd}\{n,d_1-d_2,d_1-d_3,\dots , d_1-d_t\}$ .

Proof Let $p=p(\Gamma )$ and $r=\mathrm {gcd}\{n,d_1-d_2,d_1-d_3,\dots , d_1-d_t\}$ . Observe that

$$\begin{align*}(n-d_2)d_1+d_1d_2+0d_3+\dots + 0d_t \equiv 0 \bmod n, \end{align*}$$

so there is a closed walk with $(n-d_2)$ arcs corresponding to the generator $d_1$ and $d_1$ arcs corresponding to the generator $d_2$ and 0 arcs corresponding to the remaining generators (and so has length $n+d_1-d_2$ ). Similarly, there are closed walks of lengths $n+d_1-d_3, \dots , n+d_1-d_t$ . Also, there is a closed walk of length n, consisting of $d_1$ arcs. Therefore, p divides $\mathrm {gcd}\{ n,n+d_1-d_2,n+d_1-d_3,\dots , n+d_1-d_t\}=r$ .

Consider a cycle of length l which has $l_i$ arcs corresponding to the generator $d_i$ , for each $1\leq i\leq t$ . Then $l=l_1+\dots + l_t$ and $\sum _{j=1}^t l_jd_j\equiv 0\bmod n$ , and hence $\sum _{j=1}^t l_jd_j\equiv 0\bmod r$ , so (since $d_j\equiv d_1 \bmod r$ for each $1\leq j\leq t$ ) we have $\sum _{j=1}^t l_jd_1\equiv 0\bmod r$ . That is, $l d_1\equiv 0 \bmod r$ . Now, if $\delta |r$ and $\delta |d_1$ , then $\delta |d_2,\dots ,d_t$ so $\delta |(n,d_1,\dots ,d_t)=1$ , so $\delta =1$ . Thus, $\mathrm {gcd}\{r,d_1\}=1$ , and so $l\equiv 0 \bmod r$ . Hence, r divides the length of any cycle in $\Gamma $ , and so r divides p. Hence $p=r$ .

Noting that $\mathrm {circ}_n\{d_1,\dots ,d_t\}$ is digon-free and triangle-free if and only if ${d_i + d_j \not \equiv 0}$ and $d_i+ d_j + d_k \not \equiv 0 \bmod n$ for each $1\leq i,j,k\leq t$ and that circulant digraphs are Cayley digraphs, and hence strong, we obtain the following.

Corollary 5.4 Let $\mathrm {circ}_n\{d_1,\dots ,d_t\}$ , where $n\geq 4$ and $1\leq d_i<n$ for each $1\leq i\leq t$ , and where $\{d_1,\dots ,d_t\}$ is a generating set for $\mathbb {Z}_n$ and suppose $d_i + d_j \not \equiv 0$ and $d_i+ d_j + d_k \not \equiv 0 \bmod n$ for each $1\leq i,j,k\leq t$ , and let $p=\mathrm {gcd}\{n,d_1-d_2,d_1-d_3,\dots , d_1-d_t\}$ . Let $R(a,b)$ be a cyclically reduced word that involves both a and b and which has exponent sums $\alpha ,\beta $ in a and $b^{-1}$ , respectively, and let $K=\langle {a,b}\ |\ {R(a,b)} \rangle $ . Then ${G=G_{\Gamma} (R)}$ is finite if and only if $\alpha \neq 0, \beta \neq 0$ , $\mathrm {gcd}\{\alpha ,\beta \}=1$ , $\alpha ^p\neq \beta ^p$ , $a^\alpha =b^\beta $ in K, in which case G is cyclic of order $|\alpha ^p-\beta ^p|$ .

5.3 Cartesian products of strong digraphs

Lemma 5.5 Let $\Gamma =\Gamma _1 \square \dots \square \Gamma _t$ be the Cartesian product of digon-free and triangle-free strong digraphs $\Gamma _1,\dots ,\Gamma _t$ with periods $p_1, \dots , p_t$ , respectively. Then $\Gamma $ is strong, digon-free, and triangle-free, with period $p(\Gamma )=\mathrm {gcd}\{p_1, \dots ,p_t\}$ .

Proof Since $\Gamma _1,\dots ,\Gamma _t$ are strong, so is $\Gamma $ [8, Theorem 10.3.2]. Since $\Gamma _1,\dots ,\Gamma _t$ are digon-free and triangle-free, so is $\Gamma $ [18, Lemma 2.4].

Let $r=\mathrm {gcd}\{p_1,\dots ,p_t\}$ and $p=p(\Gamma )$ . Since every cycle of each $\Gamma _i$ has a corresponding cycle in $\Gamma $ , the lengths of the cycles in the $\Gamma _i$ ’s are lengths of cycles in $\Gamma $ , so p divides the lengths of all cycles in all $\Gamma _i$ ’s, so p divides the greatest common divisor of these lengths. That is, p divides r.

We shall say that an arc $[ (u_1,\dots , u_t) , (v_1,\dots , v_t)]\in A(\Gamma )$ is of type r if $u_i=v_i$ for each $1\leq i \leq t$ , $i\neq r$ . Consider a cycle of length l that involves $l_r$ arcs of type r ( $1\leq r\leq t$ ). Then, for each $1\leq i\leq t$ , $l_i\equiv 0 \bmod p_i$ and so $l_i\equiv 0 \bmod r$ . Therefore, $l=l_1+\cdots + l_t\equiv 0 \bmod r$ . Thus, r divides the length of any cycle in $\Gamma $ , so l divides the greatest common divisor of the lengths of the cycles in $\Gamma $ . That is, r divides p, and hence $r=p$ .

Example 5.6 (Cartesian product of cycles)

Let $\Gamma =\Gamma _1 \square \dots \square \Gamma _t$ where $\Gamma _i$ ( $1\leq i\leq t$ ) is the cycle of length $m_i\geq 4$ ( $1\leq i\leq t$ ). Then $\Gamma $ is strong, digon-free, and triangle-free, with period $\mathrm {gcd}\{ m_1,\dots , m_t\}$ . (Note that $\Gamma $ is the Cayley digraph $\mathrm {Cay}(\mathbb {Z}_{m_1}\oplus \cdots \oplus \mathbb {Z}_{m_t},\{e_1,\dots ,e_t\})$ , where $e_1=(1,0,\dots ,0), e_2=(0,1,0,\dots ,0), \dots , e_t=(0,0,\dots ,1)$ .)

Corollary 5.7 Let $\Gamma =\Gamma _1 \square \dots \square \Gamma _t$ be the Cartesian product of digon-free and triangle-free, strong digraphs $\Gamma _1,\dots , \Gamma _t$ with periods $p_1,\dots , p_t$ , respectively, and let $p=\mathrm {gcd}\{p_1, \dots , p_t\}$ . Let $R(a,b)$ be a cyclically reduced word that involves both a and b and which has exponent sums $\alpha ,\beta $ in a and $b^{-1}$ , respectively, and let $K=\langle {a,b}\ |\ {R(a,b)} \rangle $ . Then $G=G_{\Gamma} (R)$ is finite if and only if $\alpha \neq 0, \beta \neq 0$ , $\mathrm {gcd}\{\alpha ,\beta \}=1$ , $\alpha ^p\neq \beta ^p$ , $a^\alpha =b^\beta $ in K, in which case G is cyclic of order $|\alpha ^{p}-\beta ^{p}|$ .

5.4 Direct products of strong digraphs

Lemma 5.8 Let $\Gamma =\Gamma _1 \times \dots \times \Gamma _t$ be the direct product of strong digraphs $\Gamma _1,\dots ,\Gamma _t$ with periods $p_1,\dots ,p_t$ , respectively, where $\mathrm {gcd}\{p_1,\dots , p_t\}=1$ , and where at least one of $\Gamma _1,\dots ,\Gamma _t$ is digon-free and at least one is triangle-free. Then $\Gamma $ is strong, digon-free, and triangle-free, with period $p(\Gamma )=p_1\cdots p_t$ .

Proof The digraph $\Gamma $ is strong with period $p(\Gamma )=p_1\cdots p_t$ by [14] (see [14, Theorem 1(ii) and (iii)] and page 251 and Proposition 4 of [10]; see also [8, Theorem 10.3.2]). If $\Gamma $ contains a digon (resp. triangle), then each of $\Gamma _1,\dots , \Gamma _t$ contains a digon (resp. triangle), a contradiction. Hence, $\Gamma $ is digon-free and triangle-free.

Example 5.9 (Direct product of an oriented diamond and a cycle)

Let $\Gamma = \Gamma _1\times \Gamma _2$ where $\Gamma _1$ is the oriented diamond digraph with $V(\Gamma _1)=\{u,v,w,t\}$ and $A(\Gamma _1) = \{ [u,v], [u,w], [v,t], [w,t], [t,u] \}$ , and $\Gamma _2$ is the cycle of length $n\geq 2$ where $n \not \equiv 0 \bmod 3$ . Then $\Gamma _1$ is strong and digon-free with period $3$ and $\Gamma _2$ is strong and triangle-free with period n, so $\Gamma $ is strong, digon-free, and triangle free, of period $p(\Gamma )=3n$ .

Corollary 5.10 Let $\Gamma =\Gamma _1 \times \dots \times \Gamma _t$ be the direct product of strong digraphs $\Gamma _1,\dots ,\Gamma _t$ with periods $p_1,\dots ,p_t$ , respectively, where $\mathrm {gcd}\{p_1,\dots , p_t\}=1$ , and where at least one of $\Gamma _1,\dots ,\Gamma _t$ is digon-free and at least one is triangle-free, and let $p=p_1\cdots p_t$ . Let $R(a,b)$ be a cyclically reduced word that involves both a and b and which has exponent sums $\alpha ,\beta $ in a and $b^{-1}$ , respectively, and let $K=\langle {a,b}\ |\ {R(a,b)} \rangle $ . Then $G=G_{\Gamma} (R)$ is finite if and only if $\alpha \neq 0$ , $\beta \neq 0$ , $\mathrm {gcd}\{\alpha ,\beta \}=1$ , $\alpha ^p\neq \beta ^p$ , $a^\alpha =b^\beta $ in K, in which case G is cyclic of order $|\alpha ^p-\beta ^p|$ .