1. Introduction
In geometric group theory, the class of Artin groups plays a major role, with many long-standing questions, such as decidability of the word problem, remaining open in general. One well-studied subclass of Artin groups is the class of right-angled Artin groups (RAAGs). Also known as graph groups, each RAAG is defined by a finite simplicial graph, where each edge of the graph represents a commutation relation; see [Reference Charney3] for a survey on RAAGs. The notion of a twisted Artin group was introduced in [Reference Clancy and Ellis4]; this class of groups also appears in [Reference Pride17] as a subclass of generalized Artin groups. This article explores the subclass of twisted right-angled Artin groups (T-RAAGs) as a natural extension of RAAGs.
Similar to RAAGs, the presentation of a T-RAAG is given by generators and relations: there are finitely many generators, and there is at most one relation between any two distinct generators
$a,b$
. This relation can be either a commutation
$ab = ba$
, or a so-called Klein relation
$aba = b$
. This presentation can be encoded by the use of mixed graphs, where vertices represent the generators of the group, while edges give rise to the relations. An undirected edge, connecting generators
$a$
and
$b$
, gives rise to the commutation relation
$ab = ba$
, while a directed edge with origin at
$a$
and terminus at
$b$
defines the Klein relation
$aba = b$
. If generators
$a,b$
are not connected by an edge, they are free of relations.
As basic and important examples of T-RAAGs, we have both the fundamental groups of a torus:
${\mathbb{Z}}^2 = \langle a, b \, | \, ab = ba \rangle$
, and of the Klein bottle:
$K = \langle a, b \, | \, aba = b \rangle$
. Note that
$K$
is not a RAAG because RAAGs are bi-orderable [Reference Duchamp and Krob8], but
$K$
is not. We refer to T-RAAGs also with the name twisted graph groups. More details regarding their normal forms and other properties can be found in [Reference Foniqi11, Chapter 3], and also in [Reference Foniqi12].
The focus of this paper is decision problems in T-RAAGs. We recall the three main problems in group theory here:
-
(i) Word problem: For a group
$G$
generated by a set
$X$
, does there exist an algorithm to decide whether
$w \in (X \sqcup X^{-1})^{\ast }$
represents the identity element of
$G$
? -
(ii) Conjugacy problem: For a group
$G$
generated by a set
$X$
, does there exist an algorithm to decide whether any two words
$u,v \in \left (X \sqcup X^{-1}\right )^{\ast }$
represent conjugate elements in
$G$
? -
(iii) Isomorphism problem: For some specified class
$\mathcal{C}$
of groups, does there exist an algorithm such that, for all pairs of presentations
$\langle X \mid R \rangle$
,
$\langle Y \mid S \rangle$
of groups in
$\mathcal{C}$
, the algorithm determines whether or not those presentations define isomorphic groups?
All three of these problems are decidable for RAAGs. Indeed, the first two problems are solvable in linear time by [Reference Crisp, Godelle and Wiest5], and the third problem is solvable by [Reference Droms7]. It is natural, therefore, to ask whether any of these three problems are solvable for T-RAAGs. We note that the word problem for T-RAAGs follows from the normal form theorem given in [Reference Foniqi11, Reference Foniqi12]. The isomorphism problem, however, remains wide open – see [Reference Blumer, Foniqi and Quadrelli1] for a solution in a certain subclass of ‘Droms T-RAAGs’. The following result constitutes the main contribution to this paper, in solving the second of these three decision problems.
Theorem A (Theorem3.7). The conjugacy problem is decidable in T-RAAGs.
We prove a conjugacy criterion for T-RAAGs (see Proposition3.4) that extends the well-known result for RAAGs, which is that any two cyclically reduced elements in a RAAG are conjugate if and only if they are related by a sequence of commutation relations and cyclic permutations. For T-RAAGs, extra care is required to account for Klein relations. For example, if
$aba = b$
in our T-RAAG, then
$a$
and
$a^{-1}$
represent conjugate elements, but are clearly not related by commutation relations or cyclic permutations. Our criterion is then used to construct an implementable solution to check when two elements in a T-RAAG are conjugate.
An implementable linear time solution for the conjugacy problem in RAAGs was constructed by [Reference Crisp, Godelle and Wiest5], using a construction known as pilings. This was adapted in [Reference Crowe6] to solve the twisted conjugacy problem for RAAGs in certain cases. This result can be used to prove linear time complexity for the conjugacy problem in T-RAAGs in certain cases.
Theorem B (Theorem3.13). Let
$T({\Gamma })$
be a T-RAAG with generators
$V{\Gamma } = \{x_{1}, \ldots , x_{n}, t\}$
, such that the only relations involving
$t$
are of the form
$x_{i}t = tx_{i}$
or
$x_{i}tx_{i} = t$
. Then, the conjugacy problem in
$T(\Gamma )$
can be solved in linear time.
T-RAAGs of the form described in Theorem B are cyclic extensions of RAAGs, and so the conjugacy problem for
$T({\Gamma })$
is equivalent to solving the twisted conjugacy problem in the RAAG
$A({{\Gamma } \setminus \{t\}})$
– see Subsection 3.3
We conclude the article by showing T-RAAGs are biautomatic, a property which is already known for RAAGs. There is extensive literature on automatic and biautomatic groups, and we refer the reader to [Reference Epstein9] for background information. Biautomatic groups have solvable conjugacy problem [Reference Gersten and Short13], and so the following result provides an alternative proof of Theorem A.
Theorem C (Theorem4.6). T-RAAGs are biautomatic.
The structure of this paper is as follows. Section 2 serves to establish the notation and provide the main definitions of the paper; this includes functions that help us keep track of syllable shuffling in words representing group elements of T-RAAGs. This will allow us to establish a conjugacy criterion, similar to that of graph products, which we can then use to prove our main results in Section 3. We conclude by proving biautomaticity of T-RAAGs in Section 4.
2. Preliminaries and notation
For a finite set
$A$
, denote by
$A^\ast$
the corresponding free monoid, that is, the set of all words over
$A$
, including the empty word
$\varepsilon$
. For a word
$w \in A^\ast$
, denote by
$l(w)$
its length, which is the number of letters used to write it.
For a subset
$X$
of a group, let
$X^{\pm } = X \sqcup X^{-1}$
, where
$X^{-1} = \{x^{-1} \mid x \in X \}$
. Our groups will be given by finite presentations
$G = \langle S \, | \, R \rangle$
; any element
$g \in G$
can be written as a word over
$S^{\pm }$
. We let
$\langle X \rangle$
denote the subgroup of
$G$
generated by
$X$
.
For a group
$G = \langle X \rangle$
, and words
$u,v \in (X^{\pm })^*$
, we let
$u = v$
denote equality of words,
$u =_{G} v$
denote the equality of group elements represented by
$u$
and
$v$
, and
$u \sim v$
denote that
$u$
and
$v$
represent conjugate elements of
$G$
. For a word
$w \in X^{\ast }$
, we denote by
$l(w)$
the word length of
$w$
over
$X$
. For a group element
$g$
in
$G = \langle X \rangle$
, we define the length of
$g$
, denoted
$| g |_{X}$
, to be the length of a shortest representative word for the element
$g$
over
$X^{\pm }$
, that is
$|g|_{X} = \min \{ l(w) \mid w\in (X^{\pm })^*, \; w =_G g\}$
; if
$X$
is fixed or clear from the context, we write
$|g|$
.
Definition 2.1.
A simplicial graph
$\Gamma$
is a pair
$\Gamma = (V, E)$
, where
$V = V{\Gamma }$
is a non-empty set whose elements are called vertices, and
$E = E{\Gamma } \subseteq \{\{x,y\} \mid x,y \in V{\Gamma }, x\neq y\}$
is a set of pairwise distinct vertices, whose elements are called edges.
Definition 2.2.
For a finite simplicial graph
$\Gamma = (V, E)$
, the right-angled Artin group (RAAG) based on
$\Gamma$
is
\begin{align*}A(\Gamma ) \;:\!=\; \langle \, V \mid uv = vu \textit{ whenever } \{u,v\}\in E\,\rangle . \\[-30pt] \end{align*}
2.1. Graph products
Here, we give an overview of graph products, and their normal form from [Reference Green14].
Definition 2.3.
Let
$\Gamma = (V,E)$
be a simplicial graph with
$V=\{1, 2, \ldots , n\}$
, and let
$G_i$
be groups, indexed by
$i\in V$
. The graph product of groups
$G_1, G_2, \ldots , G_n$
, based on
$\Gamma$
, is given by the presentation
\begin{equation*}G(\Gamma ) \;:\!=\; \langle G_1, G_2, \ldots , G_n \bigm | [G_i, G_j] = 1, \forall \{i, j\} \in E \rangle .\end{equation*}
One refers to the groups
$G_i$
(for
$1\leq i \leq n$
) as generating groups of
$G(\Gamma )$
.
Taking
$G_i = \mathbb{Z}$
, for all
$i \in V$
, the graph product
$G(\Gamma )$
becomes the RAAG
$A(\Gamma )$
; thus, RAAGs are special cases of graph products.
Definition 2.4.
Let
$G(\Gamma )$
be the graph product of the groups
$G_1, G_2, \ldots , G_n$
, and let
$w$
be a word in the generators of the
$G_i$
. If
$w = w_1 w_2 \ldots w_r$
, where each
$w_i$
is a word in the generators of only one of the generating groups, no
$w_i$
is the empty word, and
$w_i$
and
$w_{i+1}$
are not in the same generating group for all
$1 \leq i \leq r-1$
, then:
-
(i) the words
$w_i$
(for
$1 \leq i \leq r$
) are called the syllables of
$w$
, -
(ii)
$w_1, w_2, \ldots , w_r$
is called a sequence of syllables representing the element
$w$
, -
(iii) the syllable length
$\lambda (w)$
of
$w$
is equal to
$r$
, and
-
(iv) the syllable length
$\lambda (g)$
of an element
$g \in G(\Gamma )$
is the minimal syllable length of words representing
$g$
, that is
$\lambda (g) \;:\!=\; \min \{\lambda (w) \mid w =_{G(\Gamma )} g\}$
.
The syllable length has these basic properties:
$\lambda (1) = 0$
for the identity element
$1 \in G(\Gamma )$
;
$\lambda (g) = 1$
when
$g \neq 1$
belongs to one of the generating groups; and
$\lambda (g) \geq 2$
when
$g$
does not belong to any of the generating groups.
Definition 2.5.
Let
$w_1, w_2, \ldots , w_r$
be a sequence of syllables representing an element
$g$
of
$G(\Gamma )$
. For
$1 \leq i \lt j \leq r$
, we will say that the syllables
$w_i$
and
$w_j$
can be joined together if
$w_i$
and
$w_j$
belong to the same generating group, and for all
$k = i+1, \ldots , j-1$
one has
$w_iw_k =_{G(\Gamma )} w_kw_i$
.
In this case, one can group together
$w_i$
and
$w_j$
and present
$g$
with fewer syllables.
Definition 2.6.
A sequence
$w_1,\ldots ,w_r$
of syllables representing an element of
$G(\Gamma )$
is reduced if it is the empty sequence
$\varnothing$
, or if
$w_i\neq 1$
for all
$1 \leq i \leq r$
and no two syllables of the sequence can be joined.
Definition 2.7.
Let
$\cong$
be the equivalence on reduced sequences of syllables generated by
\begin{equation*} w_1,\ldots ,w_i,w_{i+1},\ldots ,w_r \ \cong \ w_1,\ldots ,w_{i+1},w_i,\ldots ,w_r \end{equation*}
whenever
$[w_i,w_{i+1}]=1$
, i.e., their vertex groups are adjacent in
$\Gamma$
. We refer to this relation as shuffling of syllables or syllable shuffling.
Now we can state the normal form theorem for graph products.
Theorem 2.8 (Reference Green14, Theorem 3.9). Let
$G(\Gamma )$
be a graph product of groups
$G_1, \ldots , G_n$
. Each element
$g\in G(\Gamma )$
can be uniquely expressed, up to syllable shuffling, as a product:
\begin{equation*}g = g_1g_2\cdots g_r\end{equation*}
where
$g_1, g_2, \ldots , g_r$
is a reduced sequence of syllables.
2.2. Twisted right-angled Artin groups
To define T-RAAGs, we use mixed graphs, which are similar to simplicial graphs, but allow directed edges.
Definition 2.9.
A mixed graph
$\Gamma = (V, E, D, o, t)$
consists of an underlying simplicial graph
$\overline {\Gamma } = (V, E)$
, a set of directed edges
$D \subseteq E$
, and maps
$o, t \colon D \to V$
giving the origin and terminus of edges. For each
$e = \{x, y\} \in D$
, we have
$o(e), t(e) \in \{x, y\}$
and
$o(e) \neq t(e)$
.
Notation 2.10.
Let
$\Gamma = (V, E, D, o, t)$
be a mixed graph.
-
(i) If
$e = \{a,b\} \in E \setminus D$
, we write
$e = [a, b]$
; note also that
$e = [b, a]$
. -
(ii) If
$e = \{a,b\} \in D$
, we write
$e = [o(e), t(e)\rangle$
.
Graphically, we present the respective edges
$[a,b]$
, and
$[a,b\rangle$
as:
In group presentations arising from graphs, the edge
$[a,b]$
denotes the commutation of
$a$
and
$b$
, while the edge
$[a,b\rangle$
denotes the corresponding Klein relation
$aba=b$
.
Definition 2.11.
Let
$\Gamma = (V, E)$
be a mixed graph. Define
\begin{equation*} T(\Gamma ) \;:\!=\; \langle V \; |\; ab = ba \ \textit{if } [a,b] \in E,\quad aba = b \ \textit{if } [a,b\rangle \in E \rangle . \end{equation*}
We call
$T(\Gamma )$
the twisted right-angled Artin group based on
$\Gamma$
, and
$\Gamma$
its defining graph.
Remark 2.12.
For a given mixed graph
$\Gamma = (V, E, D, o, t)$
, we denote by
$\overline {\Gamma }$
the underlying simplicial graph. We let
$A(\overline {\Gamma })$
denote the underlying RAAG of
$T(\Gamma )$
.
Notation 2.13.
For a T-RAAG
$\, T(\Gamma )$
with defining graph
${\Gamma } = (V,E)$
, we let
$S = V^{\pm }$
.
Definition 2.14.
Let
$\Gamma$
be a mixed graph. For
$v \in V\Gamma$
, we define
\begin{align*} {\textrm {Lk}}(v) = \{u \in V\Gamma \mid \{u,v\} \in E\Gamma \}. \\[-30pt] \end{align*}
For a subset
$A \subseteq V\Gamma$
, we define
${\textrm {Lk}}(A) = \cap _{v \in A} {\textrm {Lk}}(v)$
. Similarly we define
\begin{equation*} {\textrm {St}}(v) = {\textrm {Lk}}(v) \cup \{v\},\quad \textit{and}\quad {\textrm {St}}(A) = \cap _{v \in A} {\textrm {St}}(v). \end{equation*}
Definition 2.15.
Let
$T(\Gamma )$
be a T-RAAG with defining graph
$\Gamma$
. Any non-trivial element
$g \in T(\Gamma )$
can be written as a product
\begin{equation*} w = g_1 \ldots g_n \in S^* \end{equation*}
where each
$g_i = v_i^{a_i}$
for some
$v_i \in V\Gamma$
and
$a_i \in \mathbb{Z} \setminus \{0\}$
. Each
$g_i$
is called a syllable of
$w$
.
For
$1 \le i \lt j \le n$
, syllables
$g_i$
and
$g_j$
can be joined together if
$v_i = v_j$
and for every
$k \in \{i+1, \ldots , $
$j-1\}$
,
$v_k \in \mathrm{St}(v_i)$
.
A word
$w = g_1 \ldots g_n$
is reduced if it is empty or if
$g_i \neq 1$
(for all
$1 \leq i \leq n$
) and no two syllables in
$w$
can be joined.
2.3. Normal forms and syllable shuffling in T-RAAGs
In this subsection, we define notation that will allow us to work with reduced words representing elements of T-RAAGs in a similar way to that of graph products, whilst also keeping track of possible changes to powers when we apply syllable shuffling. For example, in the Klein bottle group
$K = \langle a,b \mid aba=b \rangle$
, the generators
$a$
and
$b$
can shuffle, but at the cost of reversing the power of
$a$
, since
$ab = ba^{-1}$
. We first introduce notation used similarly when working with graph products.
Definition 2.16.
Let
$g \in T(\Gamma )$
, let
$w = g_{1}\ldots g_{n} \in S^{\ast }$
be a reduced word representing
$g$
. We define the support of
$g$
in
$T(\Gamma )$
as
\begin{equation*} \mathrm{supp}(g) = \{v \in V{\Gamma } \mid \exists \; i \in \{1, \ldots n\} \; \textit{such that} \; g_{i} = v^{a} \; (a \in {\mathbb{Z}}) \}. \end{equation*}
The following definitions match those of [Reference Ferov10]. The abbreviation
$FL$
stands for first letters, and
$LL$
stands for last letters.
Definition 2.17.
Let
$g \in T({\Gamma })$
. Define
$\mathrm{FL}(g) \subseteq V{\Gamma }$
as the set of all
$v \in V{\Gamma }$
such that there exists a reduced word representing
$g$
which starts with the syllable
$v^{a}$
, for some
$a \in {\mathbb{Z}} \setminus \{0\}$
. Similarly define
$\mathrm{LL}(g) \subseteq V{\Gamma }$
as the set of all
$v \in V{\Gamma }$
such that there exists a reduced word representing
$g$
that ends with the syllable
$v^{a}$
. Note
$\mathrm{FL}(g) = \mathrm{LL}(g^{-1})$
.
Remark 2.18.
Let
$x = x_{1}\ldots x_{n}$
,
$y = y_{1}\ldots y_{m}$
be reduced words over
$S$
. Then, the product
$xy$
is reduced if and only if
$\mathrm{LL}(x) \cap \mathrm{FL}(y) = \varnothing$
.
Our aim is to define a function that keeps track of changing powers of generators in a word, when syllables shuffle past each other. We start by considering the generators themselves, before extending our map to words over the generating set.
Definition 2.19.
Let
$\Gamma$
be a mixed graph, and
$v \in V\Gamma$
. Define
$\varphi _{v} \colon {\textrm {St}}(v)^{\pm 1} \rightarrow {\textrm {St}}(v)^{\pm 1}$
, such that for
$x \in {\textrm {St}}(v)$
and
$\epsilon \in \{-1, 1\}$
, we map
\begin{align*} x^{\epsilon } & \mapsto \begin{cases} x^{\epsilon } & \textit{if } x = v, \textit{ or } [v, x] \in E\Gamma , \textit{ or } [v, x \rangle \in E\Gamma \\[3pt] x^{-\epsilon } & \textit{if } [x, v \rangle \in E\Gamma \end{cases} \end{align*}
Moreover, for any
$v \in V\Gamma$
, define
$\varphi _{v^{-1}} \;:\!=\; \varphi _{v}$
.
Lemma 2.20.
If
$a, b$
are adjacent vertices in
$\Gamma$
, then
$ba =_{T(\Gamma )} \varphi _{b}(a)\varphi _{a}(b)$
.
Proof. If
$e = [a,b] \in E\Gamma$
, then
$ba =_{T(\Gamma )} ab = \varphi _{b}(a)\varphi _{a}(b)$
. Similarly, if
$e = [a, b \rangle \in E\Gamma$
, then
$ba =_{T({\Gamma })} a^{-1}b = \varphi _{b}(a)\varphi _{a}(b)$
. Lastly, if
$e = [b,a \rangle \in E\Gamma$
, then
$ba =_{T({\Gamma })} ab^{-1} = \varphi _{b}(a)\varphi _{a}(b)$
.
The function
$\varphi _{s}$
can be extended as follows. If
$x \in {\textrm {St}}(s) \cap {\textrm {St}}(t)$
, for some
$s,t \in V{\Gamma }$
, then
$\varphi _{st}(x) = \varphi _{s} \circ \varphi _{t}(x)$
. This immediately implies that
$\varphi _{s^{2}}(x) = x$
, which coincides with the fact that if we shuffle a generator
$s$
past
$x^{\pm 1}$
and back again, then the power of
$x$
is unchanged. One also has
$\varphi _{st}(x) = \varphi _{ts}(x)$
.
We now extend
$\varphi _{s}$
with respect to words in T-RAAGs. For
$s \in V{\Gamma }$
, we set
$S_{s} \;:\!=\; ({\textrm {St}}(s))^{\pm }$
.
Definition 2.21.
Let
$s \in V{\Gamma }$
, let
$w = w_{1}\ldots w_{n} \in S_{s}^{\ast }$
be a reduced word. Define
\begin{align*} \varphi _{s} \colon S_{s}^{\ast } &\rightarrow S_{s}^{\ast } \\[3pt] w &\mapsto \varphi _{s}(w_{1})\ldots \varphi _{s}(w_{n}). \end{align*}
Composition works in a similar way as before: suppose
$s,t \in V{\Gamma }$
, and let
$S_{\{s, t\}} = S_s \cap S_t$
. Then, for any reduced word
$w \in S^{\ast }_{\{s,t\}}$
, we have
$\varphi _{st}(w) = \varphi _{s}\circ \varphi _{t}(w)$
.
For reduced words
$u,v \in S^{\ast }$
, we say
$\varphi _{u}(v)$
and
$\varphi _{v}(u)$
are defined if for all pairs
$(s,t)$
such that
$s \in \mathrm{supp}(u)$
and
$t \in \mathrm{supp}(v)$
, one has
$\{s,t\} \in E{\Gamma }$
. If this holds, then the words
$u$
and
$v$
can shuffle past each other in
$T({\Gamma })$
.
Syllable length of words and elements in T-RAAGs are defined similarly as in Definition2.4. In analogy with Definition2.7, and using Definition2.21, one has a definition of shuffling of syllables in T-RAAGs.
Definition 2.22.
Let
$\cong$
be the equivalence on reduced sequences generated by
\begin{equation*} w_1,\ldots ,w_{i-1}, w_i,w_{i+1}, w_{i+2}, \ldots , w_r \ \cong \ w_1,\ldots , w_{i-1}, \varphi _{a^{m}}(w_{i+1}), \varphi _{b^{n}}(w_i), w_{i+2}, \ldots ,w_r \end{equation*}
whenever
$w_i = a^m$
,
$w_{i+1} = b^n$
, with
$\{a, b\} \in E\Gamma$
, for some
$m,n \in {\mathbb{Z}}_{\neq 0}$
. We refer to this relation as shuffling of syllables.
In particular, shuffling of syllables gives
$a^m b^n \longleftrightarrow b^n a^m$
and
$a^m b^n \longleftrightarrow b^n a^{(-1)^n m}$
for any
$m,n \in {\mathbb{Z}}$
, arising from
$ab = ba$
, and
$aba = b$
, respectively.
Theorem 2.23.
(Normal form theorem, Reference Foniqi11). Every element
$g \in T(\Gamma )$
can be represented by a reduced word
$g = g_{1}\ldots g_{n}$
, where each
$g_{i} = v_{i}^{a_{i}}$
for some
$v_{i} \in V{\Gamma }$
,
$a_{i} \in {\mathbb{Z}}$
.
Moreover, if we have two reduced words
$x, y \in S^{\ast }$
representing
$g$
, then we can obtain one word from the other via a finite sequence of syllable shuffling. In particular,
$\lambda (x) = \lambda (y)$
.
We return to our map
$\varphi _{s}$
and show this function is well defined in T-RAAGs, as a consequence of the normal form theorem.
Remark 2.24.
Note that for all reduced words
$u,v \in S^{\ast }$
, we have that
$\varphi _{uv} = \varphi _{vu}$
.
Lemma 2.25.
The map
$\varphi _{s}$
is well defined for group elements in
$T(\Gamma )$
. In particular, if
$u, x, y \in S^{\ast }$
are reduced words such that
$x =_{T({\Gamma })} y$
and
$\varphi _{u}(x)$
,
$\varphi _{u}(y)$
are defined, then
\begin{equation*} \varphi _{u}(x) =_{T({\Gamma })} \varphi _{u}(y). \end{equation*}
Proof. By Theorem2.23,
$x$
and
$y$
are related to each other by syllable shuffling only. Therefore, it is enough to show that this result holds for words
\begin{equation*} x = gh, \quad y = \varphi _{g}(h)\varphi _{h}(g), \end{equation*}
for some syllables
$g = v_{i}^{a_{i}}$
,
$h = v_{j}^{a_{j}}$
such that
$\{v_{i}, v_{j}\} \in E{\Gamma }$
. We have
\begin{align*} \varphi _{u}(x) &= \varphi _{u}(g)\varphi _{u}(h) \\[3pt] &=_{T({\Gamma })} \varphi _{g}(\varphi _{u}(h))\varphi _{h}(\varphi _{u}(g)) \\[3pt] &= \varphi _{gu}(h)\varphi _{hu}(g) \\[3pt] &= \varphi _{ug}(h)\varphi _{uh}(g) \\[3pt] &= \varphi _{u}(\varphi _{g}(h))\varphi _{u}(\varphi _{h}(g)) = \varphi _{u}(y), \end{align*}
where in the fourth line we use Remark2.24.
Using this function, we can define the following move which will be useful in understanding conjugacy in T-RAAGs.
Definition 2.26.
Let
$s \in V{\Gamma }$
, let
$w = w_{1}\ldots w_{n} \in S_{s}^{\ast }$
be reduced. Define a full
$\varphi _{s}$
-cyclic permutation of
$w$
to be the reduced word
\begin{equation*} w' = \varphi _{s}(w) = \varphi _{s}(w_{1}) \ldots \varphi _{s}(w_{n}). \end{equation*}
Remark 2.27.
Whilst this may not appear similar to a cyclic permutation of
$w$
, this definition is motivated by the definition of twisted cyclic permutations given in [Reference Crowe6
, Section 4.2].
Lemma 2.28.
Let
$w, g \in S^{\ast }$
such that
$\varphi _{w}(g),$
$\varphi _{g}(w)$
are defined. Then,
\begin{equation*} \varphi _{w}(g) = g \; \Rightarrow \; w \sim \varphi _{g}(w). \end{equation*}
Proof. We have that
\begin{align*} gwg^{-1} =_{T({\Gamma })} \varphi _{g}(w)\varphi _{w}(g)g^{-1} = \varphi _{g}(w)gg^{-1} =_{T({\Gamma })} \varphi _{g}(w). \\[-30pt] \end{align*}
Note the reverse implication does not necessarily hold: for example, in
$\langle a,b \, | \, aba=b \rangle$
, one has
$a ^{-1} = \varphi _{b}(a) \neq a$
, but
$b \sim b = \varphi _{a}(b)$
.
Definition 2.29.
Let
$u,v \in S^{\ast }$
. We say
$u$
is a full conjugate preserving (FCP)
$\varphi$
-cyclic permutation of
$v$
if there exists
$s \in V{\Gamma }$
such that
$u = \varphi _{s}(v)$
and
$\varphi _{v}(s) = s$
. In particular, in this case, Lemma
2.28
implies that
$u \sim v$
.
3. Conjugacy problem in T-RAAGs
The aim of this section is to prove Theorem3.7. We will adapt a conjugacy criteria for graph products given by [Reference Ferov10, Lemma 3.12], taking into account the necessary operation of FCP
$\varphi$
-cyclic permutations, which can occur in T-RAAGs whilst preserving conjugacy classes.
3.1. Cyclic reduction
We again establish definitions for T-RAAGs similar to that of graph products given by [Reference Ferov10].
Definition 3.1.
Let
$T(\Gamma ) = \langle S \rangle$
be a T-RAAG. Let
$w = g_{1}\ldots g_{n} \in S^{\ast }$
be a reduced word representing
$g \in T({\Gamma })$
. Call the word
$w' = g_{i+1}\ldots g_{n}g_{1}\ldots g_{i}$
, where
$i \in \{1, \ldots , n-1\}$
, a cyclic permutation of
$w$
. We say the element
$g' \in T({\Gamma })$
is a cyclic permutation of
$g$
if
$g'$
can be represented by a cyclic permutation of some reduced word representing
$g$
.
We say
$w$
is cyclically reduced if all cyclic permutations of
$w$
are reduced. We say an element
$g \in T({\Gamma })$
is cyclically reduced if either
$g = Id$
, or there exists at least one reduced word representing
$g$
, which is cyclically reduced.
The proof of the following result follows similarly to [Reference Ferov10, Lemma 3.8], which we include here for completeness.
Lemma 3.2.
Let
$g \in T(\Gamma )$
, let
$w = g_{1}\ldots g_{n} \in S^{\ast }$
be a reduced word representing
$g$
. If
$w$
is cyclically reduced, then all reduced words representing
$g$
are cyclically reduced.
Proof. The idea is to show that being cyclically reduced is preserved under shuffling. Suppose
$w$
is cyclically reduced. Let
$i \in \{1, \ldots n-1\}$
such that
$(g_{i}, g_{i+1})$
can shuffle, that is
\begin{equation*} g_{i}g_{i+1} =_{T({\Gamma })} \varphi _{g_{i}}(g_{i+1})\varphi _{g_{i+1}}(g_{i}). \end{equation*}
For notation, we will write
$\varphi _{g_{i}} (g_{i+1}) = g'_{i+1}$
and
$\varphi _{g_{i+1}}(g_{i}) = g'_{i}$
. Consider the word
\begin{equation*} w' = g_{1}\ldots g_{i-1}g'_{i+1}g'_{i}g_{i+2}\ldots g_{n} =_{T({\Gamma })} w. \end{equation*}
Let
$w'' \in S^{\ast }$
be a cyclic permutation of
$w'$
. We have three cases to check:
-
(i)
$w'' = g_{j+1}\ldots g_{i-1}g'_{i+1}g'_{i}g_{i+2}\ldots g_{n}g_{1}\ldots g_{j}$
, where
$j \lt i$
, -
(ii)
$w'' = g'_{i}g_{i+2}\ldots g_{n}g_{1}\ldots g_{i-1}g'_{i+1}$
, and -
(iii)
$w'' = g_{j+1}\ldots g_{n}g_{1}\ldots g_{i-1}g'_{i+1}g'_{i}g_{i+2}\ldots g_{j}$
, where
$j \gt i$
.
For
$(i)$
,
$w''$
can be rewritten as
\begin{equation*} v = g_{j+1}\ldots g_{i-1}g_{i}g_{i+1}g_{i+2}\ldots g_{n}g_{1}\ldots g_{j}, \end{equation*}
which is a cyclic permutation of
$w$
. Therefore,
$v$
is reduced by assumption, and so
$w''$
must also be reduced by Theorem2.23. The case
$(iii)$
can be proven with the same argument.
For
$(ii)$
, we first note that by assumption the subword
$g_{i+2}\ldots g_{n}g_{1}\ldots g_{i-1}$
is reduced. Suppose
$w''$
is not reduced. First, suppose that the syllable
$g'_{i}$
can be joined with a syllable
$g_{k}$
, for some
$k \in \{i+2, \ldots , n\}$
after shuffling. If this were true, then the syllable
$g_{i}$
and
$g_{k}$
could have been joined in
$w$
, which contradicts our assumption that
$w$
is reduced. Similarly suppose
$g'_{i}$
can be joined with a syllable
$g_{l}$
for some
$l \in \{1,\ldots , i-1\}$
. Consider the word
\begin{equation*} p = g_{i}g_{i+1}\ldots g_{n}g_{1}\ldots g_{i-1}. \end{equation*}
Our assumption would imply that the syllables
$g_{i}$
and
$g_{l}$
can be joined in
$p$
, which is a cyclic permutation of
$w$
. Since
$w$
is cyclically reduced,
$p$
is reduced, which again gives a contradiction. A symmetric argument implies that the syllable
$g'_{i+1}$
cannot be joined with any syllable
$g_{s}$
, where
$s \in \{1, \ldots i-1, i+2, \ldots n\}$
. Also the syllables
$g'_{i}$
and
$g'_{i+1}$
cannot be joined with each other, since otherwise
$g_{i}$
and
$g_{i+1}$
could be joined in
$w$
, which contradicts our assumption that
$w$
is reduced. Therefore,
$w''$
must be reduced.
We have shown that being cyclically reduced is preserved under syllable shuffling. By Theorem2.23, this implies that all reduced words representing
$g$
are cyclically reduced.
We can now state the general form for cyclically reduced elements, which will be used in the first step of our conjugacy problem algorithm later. Note that this form of cyclically reduced elements matches that of RAAGs.
Corollary 3.3.
Let
$w \in S^*$
be a reduced word representing an element
$g \in T(\Gamma )$
. Then,
$g$
is cyclically reduced if and only if
$w \not =_{T({\Gamma })} uvu^{-1}$
, for any
$u,v \in S^{\ast }$
where
$l(w) \gt l(v)$
and
$uvu^{-1}$
is reduced.
3.2. Conjugacy criterion for T-RAAGs
We can now prove the following conjugacy criteria in T-RAAGs, similar to that of [Reference Ferov10, Lemma 3.12], which will allow us to prove Theorem3.7.
Proposition 3.4.
Let
$x,y \in T({\Gamma })$
be cyclically reduced. Then
$x$
and
$y$
are conjugates in
$T({\Gamma })$
if and only if there exists a finite sequence
$w = w_{0}, w_{1}, \ldots ,w_{m} = w'$
of cyclically reduced words, such that
$w$
and
$w'$
represent the elements
$x$
and
$y$
, respectively, and each pair
$(w_{i}, w_{i+1})$
, for
$0 \leq i \leq m-1$
, is related via:
-
(i) syllable shuffling,
-
(ii) a cyclic permutation, or
-
(iii) a full conjugate preserving
$\varphi$
-cyclic permutation.
In particular, if
$x$
and
$y$
are conjugates, then
$|x| = |y|$
,
$\lambda (x) = \lambda (y)$
and
$\mathrm{supp}(x) = \mathrm{supp}(y)$
.
Note if
$T(\Gamma ) = A(\Gamma )$
is a RAAG, then FCP
$\varphi$
-cyclic permutations do not have any affect on words, that is
$\varphi _{s}(w) = w$
. In particular, this result coincides with the well-known fact in RAAGs that two cyclically reduced elements are conjugate if and only if they are related by a finite sequence of commutations (i.e. syllable shuffles) and cyclic permutations.
The following notation regarding the Peripheral–Star decomposition, defined in [Reference Ferov10], will be required in the proof of Proposition3.4.
Definition 3.5 (P-S decomposition). Let
$g \in T(\Gamma )$
. Define
$S(g) = {\textrm {Supp}}(g) \cap {\textrm {St}}({\textrm {Supp}}(g))$
, and
$P(g) = {\textrm {Supp}}(g) \setminus S(g)$
. One can write
$g$
uniquely as a reduced product
$g = s(g)p(g)$
, with
${\textrm {Supp}}(s(g)) = S(g)$
and
${\textrm {Supp}}(p(g)) = P(g)$
.
One has
${\textrm {FL}}(g) = S(g) \sqcup {\textrm {FL}}(p(g))$
,
${\textrm {LL}}(g) = S(g) \sqcup {\textrm {LL}}(p(g))$
, and
$S(p(g)) = \emptyset$
.
Remark 3.6.
Let
$g \in T(\Gamma )$
where
$g_{1}\ldots g_{n}$
is a reduced expression representing
$g$
. If
$v \in S(g)$
, then there exists
$g_{i} \in \langle v \rangle$
such that
$g_{i}$
shuffles with all other syllables in
$g$
. In particular,
$v \in \mathrm{FL}(g) \cap \mathrm{LL}(g)$
.
Proof of Proposition
3.4. First note that the three operations
$(i)-(iii)$
preserve conjugacy in
$T(\Gamma )$
. Hence, it remains to prove that if
$x$
and
$y$
are conjugate, they are related by a sequence of moves of the form
$(i)-(iii)$
, and that the conditions in the final sentence of the proposition hold. So now suppose
$x$
and
$y$
are conjugates, and further that
$\lambda (x) \geq \lambda (y)$
.
Let
$X \subseteq T({\Gamma })$
denote the set of all elements obtained from
$x$
via cyclic and FCP
$\varphi$
-cyclic permutations (elements
$\varphi _g(x)$
, such that
$\varphi _x(g) = g$
). In particular,
$X \subseteq x^{T(\Gamma )} = \{gxg^{-1} \mid g \in T({\Gamma })\}$
. Choose
$x' \in X$
such that the corresponding
$g' \in T(\Gamma )$
, where
$g'x'g'^{-1} = y$
, is of minimal syllable length. We will show that
$g' = 1$
.
Claim: There exists
$x'' \in X$
,
$g'' \in T(\Gamma )$
such that
$\lambda (g'') = \lambda (g')$
,
$g''x''g''^{-1} = y$
, and
$g''x''$
is reduced.
Proof of claim: We will prove the claim using induction on
$c = |\mathrm{LL}(g') \cap \mathrm{FL}(x') |$
.
Base case:
$c = 0$
. Here
$g'x'$
is reduced; the claim holds for
$g'' = g'$
and
$x'' = x'$
.
Now suppose
$c \gt 0$
and the claim holds for all
$c' \lt c$
. Let
$g_{1}\ldots g_{k}$
and
$x_{1}\ldots x_{n}$
be reduced expressions for
$g'$
and
$x'$
respectively. WLOG let
$v \in \mathrm{LL}(g') \cap \mathrm{FL}(x')$
such that
$g_{k}, x_{1} \in \langle v \rangle$
. We justify that no generality is lost in assuming that the intersecting syllables occur in the extremal positions
$g_k$
and
$x_1$
. Suppose instead that there exist indices
$1 \le j \le k$
and
$1 \le i \le n$
and a vertex
$v \in \mathrm{LL}(g') \cap \mathrm{FL}(x')$
such that
$g_j, x_i \in \langle v \rangle$
. Since
$v \in \mathrm{LL}(g')$
, the syllable
$g_j$
can be shuffled to the end of
$g'$
using finitely many syllable shuffles, without affecting reducedness or the represented element. Similarly, since
$v \in \mathrm{FL}(x')$
, the syllable
$x_i$
can be shuffled to the beginning of
$x'$
. These moves are of type
$(i)$
and preserve the hypotheses. Hence, after applying finitely many shuffles, we may assume that the syllables in
$\langle v \rangle$
occur as the final syllable of
$g'$
and the initial syllable of
$x'$
, reducing to the case
$g_k, x_1 \in \langle v \rangle$
considered above. We have
\begin{equation*} y =_{T({\Gamma })} g_{1}\ldots g_{k-1}(g_{k}x_{1})x_{2}\ldots x_{n}\cdot g^{-1}_{k}g^{-1}_{k-1}\ldots g^{-1}_{1}. \end{equation*}
Suppose
$g_{k} = x^{-1}_{1}$
. Then we would have
\begin{equation*} y =_{T({\Gamma })} g_{1}\ldots g_{k-1}x_{2}\ldots x_{n}x_{1}g^{-1}_{k-1}\ldots g^{-1}_{1}. \end{equation*}
One could then replace
$x'$
by
$x_{2}\ldots x_{n}x_{1}$
, which is a cyclic permutation of
$x'$
, and also can replace
$g'$
by
$g_{1}\ldots g_{k-1}$
. Since
$x_{2}\ldots x_{n}x_{1} \in X$
, this contradicts the minimality condition for
$g'$
. Therefore
$g_{k} \neq x^{-1}_{1}$
. Now write
\begin{equation*} y =_{T({\Gamma })} g_{1}\ldots g_{k-1}(g_{k}x_{1})x_{2}\ldots x_{n}x_1 \cdot (g_{k}x_1)^{-1}g^{-1}_{k-1}\ldots g^{-1}_{1}. \end{equation*}
If
$v \in S(x)$
, then
$g_k x g_k^{-1} \in X$
, and we could replace
$x'$
by this element while shortening the conjugator. This contradicts the choice of
$x'$
and
$g'$
. Hence,
$v \not \in S(x)$
, implying
${\textrm {LL}}(g') = {\textrm {LL}}(g'x_1)$
and
$v \not \in {\textrm {FL}}(x_2 \cdots x_nx_1)$
. Note that if
$g_i$
, with
$g_i \in \langle u \rangle$
, can be shuffled to the end of
$g'$
, then
$\{u, v\} \in E\Gamma$
. If
$w \in {\textrm {FL}}(x_2 \cdots x_nx_1) \setminus {\textrm {FL}}(x')$
then
$\{v, w\} \not \in E\Gamma$
and so
$w \not \in {\textrm {LL}}(g'x_1)$
.
From this, we can conclude that
$v \not \in {\textrm {LL}}(g'x_1) \cap {\textrm {FL}}(x_2 \cdots x_nx_1) \subseteq {\textrm {LL}}(g') \cap {\textrm {FL}}(x')$
, hence the intersection
${\textrm {LL}}(g'x_1) \cap {\textrm {FL}}(x_2 \cdots x_nx_1)$
is a proper subset of
${\textrm {LL}}(g') \cap {\textrm {FL}}(x')$
. Setting
$g'' = g'x_1$
and
$x'' = x_2\ldots x_nx_1$
, one has
$y = g''x''g''^{-1}$
; moreover,
$c' = |{\textrm {LL}}(g'x_1) \cap {\textrm {FL}}(x'')| \lt c$
. The inductive hypothesis completes the proof of the claim.
We now have that
$g''x'' = yg''$
. Since
$g''x''$
is reduced,
$\lambda (g''x'') = \lambda (g'') + \lambda (x'') = \lambda (g') + \lambda (x') = n+k$
. Also
$\lambda (yg'') \leq \lambda (y) + \lambda (g'') = m+k$
, where
$\lambda (y) = m$
. Therefore,
$\lambda (x) \leq \lambda (y)$
. By our assumption, this implies
$\lambda (x) = \lambda (y)$
, and so
$yg''$
is also reduced. We now assume
$y_{1}\ldots y_{n}$
,
$x_{1}\ldots x_{n}$
and
$g_{1}\ldots g_{k}$
are reduced representatives of
$y, x''$
and
$g''$
, respectively. We have
\begin{equation*} y_{1}\ldots y_{n} =_{T({\Gamma })} g_{1}\ldots g_{k}x_{1}\ldots x_{n}g^{-1}_{k}\ldots g^{-1}_{1}. \end{equation*}
By Theorem2.23, there exists reductions on the right-hand side after shuffling. Assume that the syllable
$g_k^{-1}$
can be joined up with
$g_j$
for some
$1 \leq j \leq k$
. This means that
$g_k$
shuffles with
$x_i$
for all
$1 \leq i \leq n$
.
If
$\varphi _{x''}(g_k) = g_k$
, then one would contradict the minimality of
$\lambda (g'')$
, as we would have
\begin{equation*} y_{1}\ldots y_{n} =_{T({\Gamma })} g_{1}\ldots g_{k-1} \varphi _{g_k}(x'') g^{-1}_{k-1}\ldots g^{-1}_{1}, \end{equation*}
and also
$\varphi _{g_k}(x'') \sim x''$
from Lemma2.28, so
$\varphi _{g_k}(x'') \in X$
. This corresponds precisely to a move of type
$(iii)$
, namely a FCP
$\varphi$
-cyclic permutation of
$x''$
.
So one must have
$\varphi _{x''}(g_k) = g_k^{-1}$
. Since
$g_k$
shuffles with every syllable of
$x'' = x_1 \ldots x_n$
, we have
\begin{equation*} x'' g_k^{-1} = g_k \varphi _{g_k}(x''). \end{equation*}
In particular, the above equality may be rewritten as
\begin{equation*} y_{1}\ldots y_{n} =_{T(\Gamma )} g_{1}\ldots g_{k-1} g_k^2 \varphi _{g_k}(x'') g^{-1}_{k-1}\ldots g^{-1}_{1}. \end{equation*}
Rearranging gives
\begin{equation*} y_1 \cdots y_n g_1 \cdots g_{k-1} =_{T({\Gamma })} g_1 \cdots g_{k-1}g_k^2 \varphi _{g_k}(x''). \end{equation*}
The right-hand side is reduced since
$g''x''$
is reduced and has syllable length
$n+k$
. The left-hand side is also reduced, since it is a subword of
$yg''$
. However, the syllable length here is
$n+k-1$
, which contradicts Theorem2.23. In particular,
$g_{k} \not \in S(x'')$
.
WLOG we assume there exists
$w \in \mathrm{LL}(x'') \cap \mathrm{LL}(g'')$
such that
$g_{k}, x_{n} \in \langle w \rangle$
. If
$x_{n} \neq g_{k}$
, we have
\begin{equation*} y_{1}\ldots y_{n}g_{1}\ldots g_{k-1} =_{T({\Gamma })} g_{1}\ldots g_{k} x_{1}\ldots x_{n-1}(x_{n}g_k^{-1}). \end{equation*}
Both sides are reduced by construction, but differ in syllable length, which contradicts Theorem2.23. Therefore,
$x_{n} = g_{k}$
and so after cancellation, we have
\begin{equation*} y_{1}\ldots y_{n} =_{T({\Gamma })} g_{1}\ldots g_{k-1} x_{n}x_{1}\ldots x_{n-1}g^{-1}_{k-1}\ldots g^{-1}_{1}. \end{equation*}
Again a contradiction, as we could replace
$x''$
with the cyclic permutation
$x_{n}x_{1}\ldots x_{n-1}$
, and replace
$g''$
with a shorter conjugator. Indeed, this contradiction always occurs unless
$g''=1$
. Therefore,
$y = x'' \in X$
, which completes the proof.
Theorem 3.7. The conjugacy problem in T-RAAGs is solvable.
Proof. Our algorithm is as follows:
Input: T-RAAG
$T({\Gamma })$
, reduced words
$u,v \in S^{\ast }$
.
Step 1: Cyclic reduction Cyclically reduce
$u$
and
$v$
, using Corollary3.3.
Step 2: Set of representatives Let
$D_{u}$
denote the set of all words obtained from
$u$
via syllable shuffles, cyclic permutations and FCP
$\varphi$
-cyclic permutations. By Proposition3.4,
$D_{u}$
is finite, and it remains to check whether
$v \in D_{u}$
. If
$v \in D_{u}$
, then Output = True. Otherwise, Output = False.
The following question remains open.
Question 3.8. What is the time complexity of the conjugacy problem in T-RAAGs?
The linear-time algorithm for RAAGs given by [Reference Crisp, Godelle and Wiest5] does not seem to extend to T-RAAGs. The machinery they use, namely pilings, does not have an analogous definition for T-RAAGs, since we lose uniqueness of piling representatives. Moreover, their algorithm relies on the fact that two words in a cyclic normal form are conjugate in a RAAG if and only if they are equal up to a cyclic permutation. For T-RAAGs, this is not necessarily true. For example, in the Klein-bottle group
$K = \langle a, b \mid aba = b \rangle$
, the words
$a$
and
$a^{-1}$
represent conjugate elements and are cyclic normal forms, but are not equal up to cyclic permutation.
3.3. Linear-time example
We conclude this section by considering a subclass of T-RAAGs with a specific defining graph, of which we can determine linear-time complexity of the conjugacy problem for the associated T-RAAG. This is due to a connection with another decision problem, known as the twisted conjugacy problem.
Definition 3.9.
Let
$G$
be a group with finite generating set
$X$
, let
$u,v \in X^{\ast }$
, and let
$\varphi \in \mathrm{Aut}(G)$
. We say
$u$
and
$v$
are
$\varphi$
-twisted conjugates if there exists
$w \in G$
such that
$v =_{G} \varphi (w)^{-1}uw$
.
The
$\varphi$
-twisted conjugacy problem for
$G$
, denoted by
$\mathrm{TCP}_{\varphi }(G)$
, takes as input two words
$u,v \in X^{\ast }$
, and decides whether these represent group elements which are
$\varphi$
-twisted conjugate in
$G$
.
The motivation for studying this decision problem comes from [Reference Bogopolski, Martino and Ventura2], which connects the
$\varphi$
-twisted conjugacy problem of a group to the conjugacy problem in group extensions.
Theorem 3.10. [Reference Bogopolski, Martino and Ventura2 , Theorem 3.1] Let
\begin{equation} 1 \rightarrow F \xrightarrow {\alpha } G \xrightarrow {\beta } H \rightarrow 1 \end{equation}
be an algorithmic short exact sequence of groups such that
-
(i)
$F$
has solvable twisted conjugacy problem,
-
(ii)
$H$
has solvable conjugacy problem, and
-
(iii) for any
$1 \neq h \in H$
, the subgroup
$\langle h \rangle$
has finite index in its centralizer
$C_{H}(h)$
, and there is an algorithm which computes a finite set of coset representatives
$z_{h,1}, \ldots z_{h, t_{h}} \in H$
, that is
\begin{equation*} C_{H}(h) = \langle h \rangle z_{h,1} \sqcup \ldots \sqcup \langle h \rangle z_{h,t_{h}}. \end{equation*}
Then, the conjugacy problem for
$G$
is decidable if and only if the action subgroup
$A_{G} = \{\varphi _{g} \mid g \in G \} \leq \mathrm{Aut}(F)$
is orbit decidable.
We first clarify the terminology of this theorem.
Definition 3.11.
Let
$A \leq \mathrm{Aut}(G)$
for a group
$G$
with finite generating set
$X$
. The orbit decidability problem for
$A$
takes as input two words
$u,v \in X^{\ast }$
and decides whether there exists
$\varphi \in A$
such that
$v \sim \varphi (u)$
.
For a short exact sequence of groups of the form
(1)
, we define the action subgroup to be the set of all automorphisms
$\varphi _{g} \in \mathrm{Aut}(F)$
induced by conjugation, where
$g \in G$
.
We note that the hypothesis
$(i)$
from Theorem3.10 can be weakened to give the same result. Indeed, it is sufficient to have a solution to
$\mathrm{TCP}_{\varphi }(F)$
for all automorphisms
$\varphi \in A_{G}$
in the action subgroup.
We will now consider a subclass of T-RAAGs, which can be viewed as a group extension of RAAGs. This subclass is constructed using a specific type of automorphisms of RAAGs called inversions. For a RAAG
$A(\Gamma )$
with generating set
$V(\Gamma )$
, an inversion maps
$x \mapsto x^{-1}$
, for some
$x \in V(\Gamma )$
, and fixes all other vertices. With respect to these types of automorphisms, we can solve the twisted conjugacy problem in any RAAG by the following:
Theorem 3.12.
[Reference Crowe6
, Theorem 4.1] The twisted conjugacy problem
$\mathrm{TCP}_{\varphi }(A_{\Gamma })$
is solvable for all RAAGs in linear time, when
$\phi \in \mathrm{Aut}(A_{\Gamma })$
is a composition of inversions.
Consider a T-RAAG
$T(\Gamma )$
with vertices
$V{\Gamma } = \{x_{1}, \ldots , x_{n}, t \}$
, and let
$\widehat {{\Gamma }}$
be the induced subgraph of
$\Gamma$
with vertices
$\{x_{1}, \ldots , x_{n}\}$
. Also, suppose
$T(\Gamma ) = A(\widehat {{\Gamma }})$
and that for all
$i \in \{1, \ldots , n\}$
,
$T({\Gamma })$
contains the relation
$[x_{i}, t]$
or
$[x_{i}, t \rangle$
. In this case,
$T({\Gamma })$
is isomorphic to a cyclic extension of the RAAG
$A(\widehat {{\Gamma }})$
. The extension is constructed via a composition of inversions, and in particular, we can apply Theorems3.10 and 3.12 to prove linear time complexity of the conjugacy problem for this subclass of T-RAAGs.
Theorem 3.13.
Let
$T(\Gamma )$
be a T-RAAG such that
\begin{equation*} T(\Gamma ) \cong A(\widehat {\Gamma }) \rtimes _{\varphi } {\mathbb{Z}} = \langle V\widehat {\Gamma }, t \mid R(A(\widehat {\Gamma })), \; txt^{-1} = \varphi (x) \; (x \in V\widehat {\Gamma }) \rangle , \end{equation*}
where
$\varphi \in \mathrm{Aut}(A(\widehat {\Gamma }))$
is a composition of inversions. Then, the conjugacy problem in
$T(\Gamma )$
can be solved in linear time.
Proof. We first consider the hypotheses
$(i)-(iii)$
from Theorem3.10.
For
$(i)$
, consider the action subgroup
$A_{T(\Gamma )}$
. The non-trivial automorphisms of this group are precisely of the form
\begin{equation*} \varphi _{t}\;:\; x \mapsto t^{-1}xt = \varphi (x) \end{equation*}
where
$\varphi \in \mathrm{Aut}(A(\widehat {\Gamma }))$
acts either by inversion or trivially on
$x$
. In particular, the action subgroup
$A_{T(\Gamma )}$
is precisely the subgroup generated by the inversions of
$A(\widehat {\Gamma })$
. Therefore,
$(i)$
holds by Theorem3.12 and the discussion above. Hypotheses
$(ii)$
and
$(iii)$
also hold since
$\mathbb{Z}$
is a free group.
It remains to check whether the action subgroup is orbit decidable. For every
$\varphi \in A_{T(\Gamma )}$
, we can compute the image
$\varphi (u)$
– this is straightforward as
$\varphi$
is a composition of inversions. Then, we can apply the linear time solution to the conjugacy problem in RAAGs from [Reference Crisp, Godelle and Wiest5] to determine if
$v \sim \varphi (u)$
. We continue this process for all elements in the action subgroup, noting that
$|A_{T(\Gamma )}| = 2^{n}$
, since
$|V\widehat {\Gamma }| = n$
, which is fixed based on the defining graph. In particular, the action subgroup is orbit decidable, with linear time complexity.
4. Biautomaticity of T-RAAGs
For our purposes, we will work with the following definition of biautomaticity given by Sarah Rees in [Reference Ohshika and Papadopoulos16, Chapter 14]. Recall, for
$G = \langle X \rangle$
, for
$k \in \mathbb{N}$
and words
$w,v \in (X^{\pm })^{\ast }$
, we say
$w$
and
$v$
$k$
-fellow travel in
$G$
, if for each
$i \leq \mathrm{max}\{\ell (w), \ell (v)\}$
, one has
$|\mathrm{pre}_{i}(w)^{-1}\mathrm{pre}_{i}(v)| \leq k$
in
$\mathsf{Cay}(G,X)$
, where
$\mathrm{pre}_{i}(w)$
denotes the prefix of length
$i$
of
$w$
. Equivalently, we say the paths
$(1,w)$
and
$(1,v)$
of
$\mathsf{Cay}(G,X)$
$k$
-fellow travel.
Definition 4.1.
Let
$G = \langle X \rangle$
. We define a language for
$G$
over
$X^{\pm }$
to be a subset of
$(X^{\pm })^{\ast }$
that contains at least one representative for each element of
$G$
. We say
$G$
is automatic if
-
(A1) there exists a regular language for
$G$
over
$X^{\pm }$
, and
-
(A2) there exists
$k \in {\mathbb{Z}}$
such that, for each
$y \in X^{\pm } \cup \{1\}$
, and for any
$w,v \in L$
with
$wy =_{G} v$
, the paths
$(1, w), (1,v)$
$k$
-fellow travel in
$\mathsf{Cay}(G,X)$
.
We say
$G$
is biautomatic if it is automatic and also satisfies the following:
-
(A3) there exists
$k \in {\mathbb{Z}}$
such that, for each
$y \in X^{\pm } \cup \{1\}$
, and for any
$w,v \in L$
with
$yw =_{G} v$
, the paths
$(1, w), (1,v)$
$k$
-fellow travel in
$\mathsf{Cay}(G,X)$
.
Biautomatic groups have solvable conjugacy problem [Reference Gersten and Short13]; however, it is unknown whether automatic groups have solvable conjugacy problem. RAAGs were first shown to be biautomatic by [Reference Van Wyk18]. This was later shown by [Reference Hermiller and Meier15] in the more general context of graph products. We take inspiration primarily from [Reference Hermiller and Meier15], to describe another method that implies biautomaticity in RAAGs, which we will then apply to twisted RAAGs.
In [Reference Hermiller and Meier15], a finite state automaton is constructed that accepts a normal form for a RAAG
$A(\Gamma )$
over the monoid generating set
$V\Gamma$
. We will adapt this automaton, which will then accept the normal form given by [Reference Foniqi12] for T-RAAGs over
$V\Gamma$
, and so T-RAAGs satisfy (A1) from Definition4.1. We will then show that (A2) and (A3) both hold in T-RAAGs, with respect to the language accepted by our new automaton.
Before building our new automaton, we briefly discuss a construction called admissible graphs, which we can then extend similar to [Reference Hermiller and Meier15].
Definition 4.2.
We say a word
$w = ijk_{1}\ldots k_{m}$
, where
$i,j, k_{l} \in V{\Gamma }$
(for
$1 \leq l \leq m$
) is a
$(i,j)$
-admissible string if
-
(i)
$i \gt j$
and
$\{i,j\} \not \in E{\Gamma }$
, -
(ii)
$j \lt k_{1} \lt k_{2} \lt \ldots \lt k_{m}$
, and
-
(iii) if
$k_{l} \leq i$
for some
$1 \leq l \leq m$
, then
$k_{l} \not \in \mathrm{Lk}(\{i,j, k_{1}, \ldots , k_{l-1}\})$
.
The corresponding
$(i,j)$
-admissible tree is the directed graph of the form
\begin{equation*} i \xrightarrow {j} j \xrightarrow {k_{1}} k_{1} \xrightarrow {k_{2}} k_{2} \rightarrow \ldots \xrightarrow {k_{n}} k_{m}. \end{equation*}
Set
$V\Gamma = \{v_{1}, \ldots , v_{n}\}$
. We define an order on the generating set
$V\Gamma ^{\pm }$
as
\begin{equation*} v_{1} \lt v_{1}^{-1} \lt v_{2} \lt v_{2}^{-1} \lt \ldots \lt v_{n} \lt v_{n}^{-1}. \end{equation*}
Our proposed automaton
$\mathcal{M}_{\Gamma }$
is constructed as follows:
-
1. Fix a start state
$Q$
, and construct
$2n$
accept states, which we label by the generators
$V_{i}, V^{-1}_{i} \in V{\Gamma }^{\pm }$
, for
$1 \leq i \leq n$
. Add edges from
$Q$
labelled
$v_{i}$
(resp.
$v^{-1}_{i}$
) to each state
$V_{i}$
(resp.
$V^{-1}_{i}$
), for all
$1 \leq i \leq n$
. Add a loop at each state
$V_{i}$
(resp.
$V^{-1}_{i})$
labelled
$v_{i}$
(resp.
$v^{-1}_{i}$
), for all
$1 \leq i \leq n$
. -
2. For each
$1 \leq i \lt j \leq n$
, add edges from states
$V_{i}$
and
$V^{-1}_{i}$
to
$V_{j}$
labelled
$v_{j}$
, and add edges from states
$V_{i}$
and
$V^{-1}_{i}$
to
$V^{-1}_{j}$
labelled
$v^{-1}_{j}$
. -
3. For all possible
$(i,j)$
-admissible trees
$T$
, attach a copy of
$T$
to the state
$V_{i}$
, and relabel the vertices of the tree as accept states
$V_{j}, V_{k_{l}}$
for
$1 \leq l \leq n$
. Similarly, attach a second copy of
$T$
to the state
$V^{-1}_{i}$
and relabel the vertices of the tree as accept states
$V^{-1}_{j}, V^{-1}_{k_{l}}$
for
$1 \leq l \leq n$
. For every edge
$(s,t) \in T$
, add an edge from
$V_{s}$
(resp.
$V^{-1}_{s}$
) to
$V^{-1}_{t}$
(resp.
$V_{t}$
) labelled
$v^{-1}_{t}$
(resp.
$v_{t}$
) (here we are doubling the admissible trees). Also, for each state
$V^{\pm 1}_{j}, V^{\pm 1}_{k_{l}}$
, add a loop labelled
$v^{\pm 1}_{j}, v^{\pm 1}_{k_{l}}$
resp. -
4. For each pair of new states
$V^{\epsilon _{i}}_{i}, V^{\epsilon _{j}}_{j}$
, where
$\epsilon _{i}, \epsilon _{j} = \pm 1$
, added from the admissible trees, such that
$v_{j} \lt v_{i}$
and
$(v_{i}, v_{j}) \not \in E{\Gamma }$
, add an edge from
$V^{\epsilon _{i}}_{i}$
to
$V^{\epsilon _{j}}_{j}$
labelled
$v^{\epsilon _{j}}_{j}$
. -
5. Fix a fail state
$F$
. If a state
$V_{i}$
(resp.
$V^{-1}_{i}$
) is missing an edge
$v^{\epsilon }_{j}$
, then add an edge from
$V_{i}$
(resp.
$V^{-1}_{i}$
) to
$F$
labelled
$v^{\epsilon }_{j}$
, where
$\epsilon = \pm 1$
.
We let
$\mathcal{M}_{\Gamma }$
denote the automaton built from this construction. By Theorem2.23, there exists a unique shortlex shortest word, with respect to the order on
$V\Gamma ^{\pm }$
given above, representing each group element of
$T(\Gamma )$
. To determine whether a word
$w = w_{1}\ldots w_{n} \in V\Gamma ^{\pm }$
, where
$w_{i} = v^{\varepsilon }_{j_{i}}$
for some
$i \in \{1, \ldots , n\}$
,
$j_{i} \in \{1, \ldots , m\}$
,
$\varepsilon = \pm 1$
, is the unique shortlex shortest representative for a group element in
$T(\Gamma )$
, we read letters from left to right, and consider the following options when we read a letter
$w_{i}$
, for
$1 \leq i \leq n$
:
-
(i) We read the first letter
$w_{1}$
, which is accepted by
$\mathcal{M}_{\Gamma }$
; this corresponds to Step 1 of our construction above. Now consider reading a letter
$w_i$
for
$1 \lt i \leq n$
. -
(ii) If
$w_{i} = w_{i-1}$
, this letter is accepted and we read the next letter of
$w$
. If
$w_{i} = w_{i-1}^{-1}$
, then
$w$
cannot be a normal form, since
$w$
is not geodesic, and we reject
$w$
. In
$\mathcal{M}_{{\Gamma }}$
, this corresponds to Step 2 and 5 of our construction. -
(iii) If
$\{v_{j_{i-1}}, v_{j_{i}}\} \not \in E{\Gamma }$
, then we accept the subword
$w_{1}\ldots w_{i}$
and read the next letter of
$w$
. Here, when we read
$w_{i}$
, we reach a state in one of the copies of the admissible trees, added in Step 3 of the construction. -
(iv) If
$\{v_{j_{i-1}}, v_{j_{i}}\} \in E{\Gamma }$
and
$v^{\pm 1}_{j_{i-1}} \gt v^{\pm 1}_{j_{i}}$
, then the subword
$w_{1}\ldots w_{i-1}w_{i}$
is no longer shortlex shortest, since it can be rewritten as
$w_{1}\ldots \varphi _{w_{i-1}}(w_{i})\varphi _{w_{i}}(w_{i-1})$
in
$T(\Gamma )$
. In this case, we reject
$w$
, which is consistent with Step 5 of the construction. -
(v) Otherwise, suppose
$\{v_{j_{i-1}}, v_{j_{i}}\} \in E{\Gamma }$
and
$v^{\pm 1}_{j_{i-1}} \lt v^{\pm 1}_{j_{i}}$
. Then, we accept
$w$
if and only if there does not exist
$k \in \{1, \ldots , i-2\}$
such that
$v^{\pm 1}_{j_{i}} \lt v^{\pm 1}_{j_{k}}$
and
$v_{j_{i}} \in \mathrm{Lk}(\{v_{j_{k+1}}, \ldots , v_{j_{i-1}}\})$
. In this case, we could rewrite
$w$
as
$w_{1}\ldots \varphi _{z}(w_{i})\varphi _{w_{i}}(w_{k}\ldots w_{i-1})$
, where
$z= w_{k}\ldots w_{i-1}$
. This corresponds to Steps 3–5 of our construction.
The following result is then immediate from the discussion above.
Corollary 4.3.
Let
$L$
be the language accepted by the automaton
$\mathcal{M}_{\Gamma }$
. Then,
$L$
is precisely the language of normal forms for
$T(\Gamma )$
over
$V\Gamma ^{\pm }$
.
In particular, T-RAAGs satisfy (A1). To show (A2) and (A3), we first prove these conditions for RAAGs, with respect to the language accepted by our automaton
$\mathcal{M}_{\Gamma }$
.
Lemma 4.4.
RAAGs satisfy (A2) and (A3) with respect to the language
$L$
accepted by the automaton
$\mathcal{M}_{\Gamma }$
.
Proof. Let
$w,v \in L$
such that
$wy =_{G} v$
for some
$y \in V\Gamma ^{\pm }$
. We can write
$w = w_{1}y^{n}w_{2}$
for some
$n \in {\mathbb{Z}}$
, such that for every syllable
$g_{s}$
in
$w_{2}$
, where
$g_{s} \in \langle v_t \rangle$
for some
$t \in \{1, \ldots , n\}$
, we have that
$(y, v_{t}) \in E\Gamma$
, and
$y \lt v^{\pm 1}_{t}$
with respect to the order on the generating set.
Hence, we can write
$v_{2} =_{G} w_{1}y^{n+1}w_{2}$
, such that
$w_{1}y^{n+1}w_{2} \in L$
. It is then immediate that
$w$
and
$v$
2-fellow travel, and so (A2) holds for RAAGs. (A3) also holds using a similar proof.
The following result will allow us to complete our proof of biautomaticity in T-RAAGs.
Theorem 4.5.
[Reference Foniqi11
, Theorem 5.9] Let
$T(\Gamma )$
be a twisted RAAG, and let
$A(\Gamma )$
be its underlying RAAG. Then, the Cayley graphs of
$T(\Gamma )$
and
$A(\Gamma )$
are isomorphic as undirected graphs, with respect to the standard generating set
$S = V\Gamma ^{\pm }$
.
In particular, distances are preserved between
$\mathsf{Cay}(A(\Gamma ), S)$
and
$\mathsf{Cay}(T(\Gamma ), S)$
, and so T-RAAGs also satisfy (A2) and (A3), using Lemma4.4. The following is then immediate, which gives an alternative positive solution to the conjugacy problem in T-RAAGs.
Theorem 4.6. T-RAAGs are biautomatic.
Acknowledgements
The authors would like to thank Yago Antolín for inviting Islam to speak at the Heriot-Watt MAXIMALs seminar, from which this collaboration began. The authors are grateful to the anonymous referee and to Marius Streiff for helpful comments and suggestions.
Islam Foniqi acknowledges support from the EPSRC Fellowship grant EP/V032003/1 ‘Algorithmic, topological and geometric aspects of infinite groups, monoids and inverse semigroups’.
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