Palindromic automorphisms of right-angled Artin groups
Neil J. Fullarton and Anne Thomas

arXiv:1510.03939v2 [] 28 Oct 2016

April 11, 2018
Abstract
We introduce the palindromic automorphism group and the palindromic Torelli
group of a right-angled Artin group AΓ . The palindromic automorphism group ΠAΓ
is related to the principal congruence subgroups of GL(n, Z) and to the hyperelliptic
mapping class group of an oriented surface, and sits inside the centraliser of a certain
hyperelliptic involution in Aut(AΓ ). We obtain finite generating sets for ΠAΓ and for
this centraliser, and determine precisely when these two groups coincide. We also find
generators for the palindromic Torelli group.

1

Introduction

Let Γ be a finite simplicial graph, with vertex set V = {v1 , . . . , vn }. Let E ⊂ V × V be the
edge set of Γ. The graph Γ defines the right-angled Artin group AΓ via the presentation
AΓ = hvi ∈ V | [vi , vj ] = 1 iff (vi , vj ) ∈ Ei.
One motivation, among many, for studying right-angled Artin groups and their automorphisms (see Agol [1] and Charney [3] for others) is that the groups AΓ and Aut(AΓ )
allow us to interpolate between families of groups that are classically well-studied: we
may pass between the free group Fn and free abelian group Zn , between their automorphism groups Aut(Fn ) and Aut(Zn ) = GL(n, Z), and even between the mapping class
group Mod(Sg ) of the oriented surface Sg of genus g and the symplectic group Sp(2g, Z)
(this last interpolation is explained in [8]). See Section 2 for background on right-angled
Artin groups and their automorphisms.
In this paper, we introduce a new subgroup of Aut(AΓ ) consisting of so-called ‘palindromic’
automorphisms of AΓ , which allows us a further interpolation, between certain previously
well-studied subgroups of Aut(Fn ) and of GL(n, Z). An automorphism α ∈ Aut(AΓ ) is said
to be palindromic if α(v) ∈ AΓ is a palindrome for each v ∈ V ; that is, each α(v) may be
expressed as a word u1 . . . uk on V ±1 such that u1 . . . uk and its reverse uk . . . u1 are identical
as words. The collection ΠAΓ of palindromic automorphisms is, a priori, only a subset of
Aut(AΓ ). While it is easy to see that ΠAΓ is closed under composition, it is not obvious
that it is closed under inversion. In Corollary 3.5 we prove that ΠAΓ is in fact a subgroup
of Aut(AΓ ). We thus refer to ΠAΓ as the palindromic automorphism group of AΓ .
When AΓ is free, the group ΠAΓ is equal to the palindromic automorphism group ΠAn
of Fn , which was introduced by Collins [5]. Collins proved that ΠAn is finitely presented and

1

provided an explicit finite presentation. The group ΠAn has also been studied by Glover–
Jensen [10], who showed, for instance, that it has virtual cohomological dimension n − 1.
At the other extreme, when AΓ is free abelian, the group ΠAΓ is the principal level 2
congruence subgroup Λn [2] of GL(n, Z). Thus ΠAΓ enables us to interpolate between these
two classes of groups.
Let ι be the automorphism of AΓ that inverts each v ∈ V . In the case that AΓ is free,
it is easy to verify that the palindromic automorphism group ΠAΓ = ΠAn is equal to the
centraliser CΓ (ι) of ι in Aut(AΓ ) (hence ΠAn is a group). For a general AΓ , we prove
that ΠAΓ is a finite index subgroup of CΓ (ι), by first considering the finite index subgroup
of ΠAΓ consisting of ‘pure’ palindromic automorphisms; see Theorem 3.3 and Corollary 3.5.
The index of ΠAΓ in CΓ (ι) depends entirely on connectivity properties of the graph Γ, and
we give conditions on Γ that are equivalent to the groups ΠAΓ and CΓ (ι) being equal, in
Proposition 3.6. In particular, there are non-free AΓ such that ΠAΓ = CΓ (ι).
The order 2 automorphism ι is the obvious analogue in Aut(AΓ ) of the hyperelliptic involution s of an oriented surface Sg , since ι and s act as −I on H1 (AΓ , Z) and H1 (Sg , Z),
respectively. The group ΠAΓ also allows us to generalise a comparison made by the first
author in [9, Section 1] between ΠAn ≤ Aut(Fn ) and the centraliser in Mod(Sg ) of the hyperelliptic involution s, which demonstrated a deep connection between these groups. Our
study of ΠAΓ is thus motivated by its appearance in both algebraic and geometric settings.
The main result of this paper finds a finite generating set for ΠAΓ . Our generating set
includes the so-called diagram automorphisms of AΓ , which are induced by graph symmetries of Γ, and the inversions ιj ∈ Aut(AΓ ), with ιj mapping vj to vj −1 and fixing every
vk ∈ V \ {vj }. The function Pij : V → AΓ sending vi to vj vi vj and vk to vk (k 6= i) induces a well-defined automorphism of AΓ , also denoted Pij , whenever certain connectivity
properties of Γ hold (see Section 3.2). We establish that these three types of palindromic
automorphisms suffice to generate ΠAΓ .
Theorem A. The group ΠAΓ is generated by the finite set of diagram automorphisms,
inversions and well-defined automorphisms Pij .
We also obtain a finite generating set for the centraliser CΓ (ι), in Corollary 3.8, by combining
the generating set given by Theorem A with a short exact sequence involving CΓ (ι) and
the pure palindromic automorphism group (see Theorem 3.3). Our generating set for CΓ (ι)
consists of the generators of ΠAΓ , along with all well-defined automorphisms of AΓ that
map vi to vi vj and fix every vk ∈ V \ {vi }, for some i 6= j with [vi , vj ] = 1 in AΓ .
Further, for any re-indexing of the vertex set V and each k = 1, . . . , n, we provide a finite generating set for the subgroup ΠAΓ (k) of ΠAΓ which fixes the vertices v1 , . . . , vk , as
recorded in Theorem 3.11. The so-called partial basis complex of AΓ , which is an analogue
of the curve complex, has as its vertices (conjugacy classes of) the images of members of V
under automorphisms of Aut(AΓ ). This complex has not, to our knowledge, appeared in the
literature, but its definition is an easy generalisation of the free group version introduced by
Day–Putman [6] in order to generate the Torelli subgroup of Aut(Fn ). A ‘palindromic’ partial basis complex was also used in [9] to approach the study of palindromic automorphisms
of Fn . Theorem 3.11 is thus a first step towards understanding stabilisers of simplices in
the palindromic partial basis complex of AΓ .
2

We prove Theorem A and our other finite generation results in Section 3, using machinery
developed by Laurence [16] for his proof that Aut(AΓ ) is finitely generated. The added
constraint for us that our automorphisms be expressed as a product of palindromic generators forces a more delicate treatment. In addition, our proof uses Servatius’ Centraliser
Theorem [18], and a generalisation to AΓ of arguments used by Collins [5, Proposition 2.2]
to generate ΠAn . Throughout this paper, we employ a decomposition into block matrices
of the image of Aut(AΓ ) in GL(n, Z) under the canonical map induced by abelianising AΓ ;
this decomposition was observed by Day [7] and by Wade [19].
We also in this work introduce the palindromic Torelli group PI Γ of AΓ , which we define to
consist of the palindromic automorphisms of AΓ that induce the identity automorphism on
H1 (AΓ ) = Zn . The group PI Γ is the right-angled Artin group analogue of the hyperelliptic
Torelli group SI g of an oriented surface Sg , which has applications to Burau kernels of braid
groups [2] and to the Torelli space quotient of the Teichmüller space of Sg [12]. Analogues
of these objects exist for right-angled Artin groups (see, for example, [4]), but are not yet
well-developed. We expect that the palindromic Torelli group will play a role in determining
their structure.
Even in the free group case, where PI Γ is denoted by PI n , little seems to be known
about the palindromic Torelli group. Collins [5] observed that PI n is non-trivial, and
Jensen–McCammond–Meier [14, Corollary 6.3] proved that PI n is not homologically finite
if n ≥ 3. An infinite generating set for PI n was obtained recently in [9, Theorem A],
and this is made up of so-called doubled commutator transvections and separating π-twists.
In Section 4 we recall and then generalise the definitions of these two classes of free group
automorphisms, to give two classes of palindromic automorphisms of a general AΓ , which
we refer to by the same names. As a first step towards understanding the structure of PI Γ ,
we obtain an explicit generating set as follows.
Theorem B. The group PI Γ is generated by the set of all well-defined doubled commutator
transvections and separating π-twists in ΠAΓ .
The generating set we obtain in Theorem B compares favourably with the generators obtained in [9] in the case that AΓ is free. Specifically, the generators given by Theorem B
are the images in Aut(AΓ ) of those generators of PI n that descend to well-defined automorphisms of AΓ (viewing AΓ as a quotient of the free group Fn on the set V ).
The proof of Theorem B in Section 4 combines our results from Section 3 with results
for PI n from [9]. More precisely, as a key step towards the proof of Theorem A, we find
a finite generating set for the pure palindromic subgroup of ΠAΓ (Theorem 3.7). We then
use these generators to determine a finite presentation for the image Θ of this subgroup
under the canonical map Aut(AΓ ) → GL(n, Z) (Theorem 4.2). In order to find this finite
presentation for Θ ≤ GL(n, Z), we also need Corollary 1.1 from [9], which leverages the
generating set for PI n from [9] to obtain a finite presentation for the principal level 2
congruence subgroup Λn [2] ≤ GL(n, Z). Finally, using a standard argument, we lift the
relators of Θ to obtain a normal generating set for PI Γ .
Acknowledgements. The authors would like to thank Tara Brendle and Stuart White,
for encouraging their collaboration on this paper. We also thank an anonymous referee for
spotting a gap in the proof of Proposition 3.1 in an earlier version, and for other helpful
comments.
3

2

Preliminaries

In this section we give definitions and some brief background on right-angled Artin groups
and their automorphisms. Throughout this section and the rest of the paper, we continue to
use the notation introduced in Section 1. We will also frequently use vi ∈ V to denote both
a vertex of the graph Γ and a generator of AΓ , and when discussing a single generator we
may omit the index i. Section 2.1 recalls definitions related to the graph Γ and Section 2.2
recalls some useful combinatorial results about words in the group AΓ . In Section 2.3 we
recall a finite generating set for Aut(AΓ ) and some important subgroups of Aut(AΓ ), and in
Section 2.4 we recall a matrix block decomposition for the image of Aut(AΓ ) in GL(n, Z).

2.1

Graph-theoretic notions

We briefly recall some graph-theoretic definitions, in particular the domination relation on
vertices of Γ.
The link of a vertex v ∈ V , denoted lk(v), consists of all vertices adjacent to v, and the star
of v ∈ V , denoted st(v), is defined to be lk(v) ∪ {v}. We define a relation ≤ on V , with
u ≤ v if and only if lk(u) ⊂ st(v). In this case, we say v dominates u, and refer to ≤ as the
domination relation [15], [16]. Figure 1 demonstrates the link of one vertex being contained
in the star of another. Note that when u ≤ v, the vertices u and v may be adjacent in Γ,
but need not be. To distinguish these two cases, we will refer to adjacent and non-adjacent
domination.

v

u

Figure 1: An example of a vertex u being dominated by a vertex v. The dashed edge is meant to
emphasise that u and v may be adjacent, but need not be.

Domination in the graph Γ may be used to define an equivalence relation ∼ on the vertex
set V , as follows. We say vi ∼ vj if and only if vi ≤ vj and vj ≤ vi , and write [vi ]
for the equivalence class of vi ∈ V under ∼. We also define an equivalence relation ∼0
by vi ∼0 vj if and only if [vi ] = [vj ] and vi vj = vj vi , writing [vi ]0 for the equivalence class
of vi ∈ V under ∼0 . We refer to [vi ] as the domination class of vi and to [vi ]0 as the adjacent
domination class of vi . Note that the vertices in [vi ] necessarily span either an edgeless or
a complete subgraph of Γ; in the former case, we will call [vi ] a free domination class, while
in the latter, where [vi ] = [vi ]0 , we will call [vi ] an abelian domination class.

4

2.2

Word combinatorics in right-angled Artin groups

In this section we recall some useful properties of words on V ±1 , which give us a measure of
control over how we express group elements of AΓ . We include the statement of Servatius’
Centraliser Theorem [18] and of a useful proposition of Laurence from [16].
First, a word on V ±1 is reduced if there is no shorter word representing the same element
of AΓ . Unless otherwise stated, we shall always use reduced words when representing
members of AΓ . Now let w and w0 be words on V ±1 . We say that w and w0 are shuffleequivalent if we can obtain one from the other via repeatedly exchanging subwords of the
form uv for vu when u and v are adjacent vertices in Γ. Hermiller–Meier [13] proved
that two reduced words w and w0 are equal in AΓ if and only if w and w0 are shuffleequivalent, and also showed that any word can be made reduced by a sequence of these
shuffles and cancellations of subwords of the form u u− (u ∈ V ,  ∈ {±1}). This allows us
to define the length of a group element w ∈ AΓ to be the number of letters in a reduced word
representing w, and the support of w ∈ AΓ , denoted supp(w), to be the set of vertices v ∈ V
such that v or v −1 appears in a reduced word representing w. We say w ∈ AΓ is cyclically
reduced if it cannot be written in reduced form as vw0 v −1 , for some v ∈ V ±1 , w0 ∈ AΓ .
Servatius [18, Section III] analysed centralisers of elements in arbitrary AΓ , showing that
the centraliser of any w ∈ AΓ is again a (well-defined) right-angled Artin group, say A∆ .
Laurence [16] defined the rank of w ∈ AΓ to be the number of vertices in the graph ∆
defining A∆ . We denote the rank of w ∈ AΓ by rk(w).
In order to state his theorem on centralisers in AΓ , Servatius [18] introduced a canonical
form for any cyclically reduced w ∈ AΓ , which Laurence [16] calls a basic form of w. For
this, partition the support of w into its connected components in Γc , the complement graph
of Γ, writing
supp(w) = V1 t · · · t Vk ,
where each Vi is such a connected component. Then we write
w = w1 r1 . . . wk rk ,
where each ri ∈ Z and each wi ∈ hVi i is not a proper power in AΓ (that is, each |ri | is
maximal). Note that by construction, [wi , wj ] = 1 for 1 ≤ i < j ≤ k. Thus the basic form
of w is unique up to permuting the order of the wi , and shuffling within each wi . With this
terminology in place, we now state Servatius’ ‘Centraliser Theorem’ for later use.
Theorem 2.1 (Servatius, [18]). Let w be a cyclically-reduced word on V ±1 representing
an element of AΓ . Writing w = w1 r1 . . . wk rk in basic form, the centraliser of w in AΓ is
isomorphic to
hw1 i × · · · × hwk i × hlk(w)i,
where lk(w) denotes the subset of V of vertices which are adjacent to each vertex in supp(w).
We will also make frequent use of the following result, due to Laurence [16], and so state it
now for reference.
Proposition 2.2 (Proposition 3.5, Laurence [16]). Let w ∈ AΓ be cyclically reduced, and
write w = w1 r1 . . . wk rk in basic form, with Vi := supp(wi ). Then:
5

1. rk(v) ≥ rk(w) for all v ∈ supp(w); and
2. if rk(v) = rk(w) for some v ∈ Vi , then:
(a) v ≤ u for all u ∈ supp(w);
(b) each Vj is a singleton (j 6= i); and
(c) v does not commute with any vertex of Vi \ {v}.
Recall that a clique in a graph Γ is a complete subgraph. If ∆ is a clique in Γ then A∆ is
free abelian of rank equal to the number of vertices of ∆, so any word supported on ∆ can
be written in only finitely many reduced ways. The set of cliques in ∆ is partially ordered
by inclusion, giving rise to the notion of a maximal clique in a graph Γ.

2.3

Automorphisms of right-angled Artin groups

In this section we recall a finite generating set for Aut(AΓ ). This generating set was obtained
by Laurence [16], confirming a conjecture of Servatius [18], who had verified that the set
generates Aut(AΓ ) in certain special cases.
In the following list, the action of each generator of Aut(AΓ ) is given on v ∈ V , with the
convention that if a vertex is omitted from discussion, it is fixed by the automorphism.
There are four types of generators:
1. Diagram automorphisms φ: each φ ∈ Aut(Γ) induces an automorphism of AΓ , which
we also denote by φ, mapping v ∈ V to φ(v).
2. Inversions ιj : for each vj ∈ V , ιj maps vj to vj −1 .
3. Dominated transvections τij : for vi , vj ∈ V , whenever vi is dominated by vj , there
is an automorphism τij mapping vi to vi vj . We refer to a (well-defined) dominated
transvection τij as an adjacent transvection if [vi , vj ] = 1; otherwise, we say τij is a
non-adjacent transvection.
4. Partial conjugations γi,D : fix vi ∈ V , and select a connected component D of Γ \ st(vi )
(see Figure 2). The partial conjugation γvi ,D maps every d ∈ D to vi dvi −1 .
We denote by DΓ , IΓ and PC(AΓ ) the subgroups of Aut(AΓ ) generated by diagram automorphisms, inversions and partial conjugations, respectively, and by Aut0 (AΓ ) the subgroup
of Aut(AΓ ) generated by all inversions, dominated transvections and partial conjugations.

2.4

A matrix block decomposition

Now we recall a useful decomposition into block matrices of an image of Aut(AΓ ) inside
GL(n, Z). This decomposition was observed by Day [7] and by Wade [19].
Let Φ : Aut(AΓ ) → GL(n, Z) be the canonical homomorphism induced by abelianising AΓ .
Note that since DΓ normalises Aut0 (AΓ ), any φ ∈ Aut(AΓ ) may be written (non-uniquely,
in general), as φ = δβ, where δ ∈ DΓ and β ∈ Aut0 (AΓ ).
6

D0

v

D00

D

Figure 2: When we remove the star of v, we leave three connected components D, D0 and D00 .

By ordering the vertices of Γ appropriately, matrices in Φ(Aut0 (AΓ )) ≤ GL(n, Z) will have
a particularly tractable lower block-triangular decomposition, which we now describe. The
domination relation ≤ on V descends to a partial order, also denoted ≤, on the set of
domination classes V / ∼, which we (arbitrarily) extend to a total order,
[u1 ] < · · · < [uk ]
where [ui ] ∈ V / ∼. This total order may be lifted back up to V by specifying an arbitrary
total order on each domination class [ui ] ∈ V / ∼. We reindex the vertices of Γ if necessary
so that the ordering v1 , v2 , . . . , vn is this specified total order on V . Let ni denote the size
of the domination class [ui ] ∈ V / ∼. Under this ordering, any matrix M ∈ Φ(Aut0 (AΓ ))
has block decomposition:


M1 0
0 ...
0
 ∗ M2 0 . . .
0 


 ∗
∗ M3 . . .
0 

,
 ..
..
..
.. 
..
 .
.
.
.
. 
∗

∗

∗

...

Mk

where Mi ∈ GL(ni , Z) and the (i, j) block ∗ (j < i) may only be non-zero if uj is dominated by ui in Γ. This triangular decomposition becomes apparent when the images of the
generators of Aut0 (AΓ ) are considered inside GL(n, Z). The diagonal blocks may be any
Mi ∈ GL(ni , Z), as by definition each domination class gives rise to all ni (ni − 1) transvections in GL(ni , Z), which, together with the appropriate inversions, generate GL(ni , Z).
A diagonal block corresponding to a free domination class will also be called free, and a
diagonal block corresponding to an abelian domination class will be called abelian.
This block decomposition descends to an analogous decomposition of the image of Aut0 (AΓ )
under the canonical map Φ2 to GL(n, Z/2), as this map factors through the homomorphism
GL(n, Z) → GL(n, Z/2) that reduces matrix entries mod 2.

3

Palindromic automorphisms

Our main goal in this section is to prove Theorem A, which gives a finite generating set for
the group of palindromic automorphisms ΠAΓ . First of all, in Section 3.1, we derive a normal
7

form for group elements α(v) ∈ AΓ where v ∈ V and α lies in the centraliser CΓ (ι). In
Section 3.2 we introduce the pure palindromic automorphisms PΠAΓ , and prove that PΠAΓ
is a group by showing that it is a kernel inside CΓ (ι). We then show that ΠAΓ is a group,
and determine when the groups CΓ (ι) and ΠAΓ are equal. The proof of Theorem A is
carried out in Section 3.3, where the main step is to find a finite generating set for PΠAΓ .
We also provide finite generating sets for CΓ (ι) and for certain stabiliser subgroups of ΠAΓ .

3.1

The centraliser CΓ (ι) and a clique-palindromic normal form

In this section we prove Proposition 3.1, which provides a normal form for reduced words
w = u1 . . . uk (ui ∈ V ±1 ) that are equal (in the group AΓ ) to their reverse,
wrev := uk . . . u1 .
We then in Corollary 3.2 derive implications for the diagonal blocks in the matrix decomposition discussed in Section 2.4. The results of this section will be used in Section 3.2
below.
Green, in her thesis [11], established a normal form for elements of AΓ , by iterating an
algorithm that takes a word w0 on V ±1 and rewrites it as w0 = pw1 in AΓ , where p is a
word consisting of all the letters of w0 that may be shuffled (as in Section 2.2) to be the
initial letter of w0 , and w1 is the word remaining after shuffling each of these letters into
the initial segment p. We now use a similar idea for palindromes.
Let ι denote the automorphism of AΓ that inverts each v ∈ V . We refer to ι as the (preferred)
hyperelliptic involution of AΓ . Denote by CΓ (ι) the centraliser in Aut(AΓ ) of ι. Note that
this centraliser is far from trivial: it contains all diagram automorphisms, inversions and
adjacent transvections in Aut(AΓ ), and also contains all palindromic automorphisms. The
following proposition gives a normal form for the image of v ∈ V under the action of
some α ∈ CΓ (ι).
Proposition 3.1 (Clique-palindromic normal form). Let α ∈ CΓ (ι) and v ∈ V . Then we
may write
α(v) = w1 . . . wk−1 wk wk−1 . . . w1 ,
where wi is a word supported on a clique in Γ (1 ≤ i ≤ k), and if k ≥ 3 then [wi , wi+1 ] 6= 1
(1 ≤ i ≤ k − 2). Moreover, this expression for α(v) is unique up to the finitely many
rewritings of each word wi in AΓ .
We refer to this normal form as clique-palindromic because the words under consideration,
while equal to their reverses in the group AΓ as genuine palindromes are, need only be
palindromic ‘up to cliques’, as in the expression in the statement of the proposition.
Proof. Suppose α ∈ CΓ (ι) and v ∈ V . Write α(v) = u1 . . . ur in reduced form, where each
ui is in V ±1 . Since αι(v) = ια(v), we have that
u1 . . . ur = ur . . . u1

(1)

in AΓ . If α(v) is supported on a clique, then there is nothing to show. Otherwise, put
A1 = α(v) and let Z1 be the (possibly empty) subset of V consisting of the vertices in
8

supp(A1 ) which commute with every vertex in supp(A1 ). We note that Z1 is supported on
a clique, and that Z1 is, by assumption, a proper subset of supp(A1 ).
We now rewrite A1 = u1 . . . ur as w1 u1 0 . . . us 0 , where uj 0 ∈ V ±1 (1 ≤ j ≤ s), and w1 ∈ AΓ
is the word consisting of all the ui which are not in Z1±1 and which may be shuffled to the
start of u1 . . . ur . That is, w1 consists of all letters ui 6∈ Z1±1 so that if i ≥ 1, the letter ui
commutes with each of u1 , . . . , ui−1 . Notice that w1 is nonempty since the first ui which is
not in Z1 will be in w1 . By construction, w1 is supported on a clique in Γ.
Now any ui that may be shuffled to the start of u1 . . . ur may also be shuffled to the end
of ur . . . u1 , by (1). Hence we may also rewrite A1 as u001 . . . u00s w1 for the same word w1 .
Since the support of w1 is disjoint from Z1 , the letters of A1 used in the copy of w1 at the
start of w1 u1 0 . . . us 0 are disjoint from the letters of A1 used in the copy of w1 at the end of
u001 . . . u00s w1 . We thus obtain that
A1 = α(v) = w1 u1 00 . . . ut 00 w1
in AΓ , with ui 00 ∈ V ±1 . Since αι(v) = ια(v), it must be the case that u1 00 . . . ut 00 = ut 00 . . . u1 00
in AΓ .
Now put A2 = u1 00 . . . ut 00 , so that A1 = w1 A2 w1 . Note that supp(A2 ) contains Z1 . If A2 is
supported on a clique, for example if supp(A2 ) = Z1 , then we put w2 = A2 and are done.
(In this case, supp(A2 ) = Z1 if and only if w1 and w2 commute.) If A2 is not supported
on a clique, we define Z2 to be the vertices in supp(A2 ) which commute with the entire
support of A2 , and iterate the process described above. Since each word wi constructed by
this process is nonempty, the word Ai+1 is shorter than Ai , hence the process terminates
after finitely many steps. Notice also that Z1 ⊆ Z2 ⊆ · · · ⊆ Zi ⊆ supp(Ai+1 ), so any letters
of Ai which lie in Zi become part of the word Ai+1 . In particular, any letter of A1 = α(v)
which is in some Zi , for example a letter in Z(AΓ ), will end up in the word wk when the
process terminates.
By construction, each wi is supported on a clique in Γ. Now the word Ai+1 is not supported
on a clique if and only if a further iteration is needed, which occurs if and only if i ≤ k − 2.
In this case, Zi must be a proper subset of supp(Ai+1 ) and so wi+1 does not commute with
wi (the word wk may or may not commute with wk−1 ). Thus the expression obtained for
α(v) when this process terminates is as in the statement of the proposition. Moreover, this
expression is unique up to rewriting each of the wi , as they were defined in a canonical
manner. This completes the proof.
This normal form gives us the following corollary regarding the structure of diagonal blocks
in the lower block-triangular decomposition of the image of α ∈ CΓ (ι) under the canonical
map Φ : Aut(AΓ ) → GL(n, Z), discussed in Section 2.4. Recall that Λk [2] denotes the
principal level 2 congruence subgroup of GL(k, Z).
Corollary 3.2. Write α ∈ CΓ (ι) as α = δβ, for some β ∈ Aut0 (AΓ ) and δ ∈ DΓ . Let M be
the matrix appearing in a diagonal block of rank k in the lower block-triangular decomposition
of Φ(β) ∈ GL(n, Z). Then:
1. if the diagonal block is abelian, then M may be any matrix in GL(k, Z); and
9

2. if the diagonal block is free then M must lie in Λk [2], up to permuting columns.
Proof. First, note that since DΓ ≤ CΓ (ι), we must have that β ∈ CΓ (ι). We deal with the
abelian block case first. The group CΓ (ι) ∩ Aut0 (AΓ ) contains all the adjacent transvections
and inversions necessary to generate GL(k, Z) under Φ, so the matrix M in this diagonal
block may be any member of GL(k, Z).
Now, suppose that the diagonal block is free. Suppose the column of M corresponding to
v ∈ V contains two odd entries, in turn corresponding to vertices u1 , u2 ∈ [v], say. This
implies that β(v) has odd exponent sum of u1 and of u2 . Use Proposition 3.1 to write
β(v) = w1 . . . wk . . . w1
in normal form, with each wi ∈ AΓ being supported on some clique in Γ. It must be the
case that wk has odd exponent sum of u1 and of u2 , since all other wi (i 6= k) appear twice
in the normal form expression. Thus u1 and u2 commute. This contradicts the assumption
that the diagonal block is free, so there must be precisely one odd entry in each column
of M . Hence up to permuting columns, we have M ∈ Λk [2].

3.2

Pure palindromic automorphisms

In this section we introduce the pure palindromic automorphisms PΠAΓ , which we will see
form an important finite index subgroup of ΠAΓ . In Theorem 3.3 we prove that PΠAΓ is a
group, by showing that it is the kernel of the map from the centraliser CΓ (ι) to GL(n, Z/2)
induced by mod 2 abelianisation. Proposition 3.4 then says that any element of ΠAΓ can
be expressed as a product of an element of PΠAΓ with a diagram automorphism, and as
Corollary 3.5 we obtain that the collection of palindromic automorphisms ΠAΓ is in fact a
group. This section concludes by establishing a necessary and sufficient condition on the
graph Γ for the groups ΠAΓ and CΓ (ι) to be equal, in Proposition 3.6.
We define PΠAΓ ⊂ ΠAΓ be the subset of palindromic automorphisms of AΓ such that for
each v ∈ V , the word α(v) may be expressed as a palindrome whose middle letter is either v
or v −1 . For instance, IΓ ⊂ PΠAΓ but DΓ ∩PΠAΓ is trivial. If vi ≤ vj , there is a well-defined
pure palindromic automorphism Pij := (ιτij )2 , which sends vi to vj vi vj and fixes every other
vertex in V . We refer to Pij as a dominated elementary palindromic automorphism of AΓ .
The following theorem shows that PΠAΓ is a group, by establishing that it is a kernel
inside CΓ (ι). We will thus refer to PΠAΓ as the pure palindromic automorphism group
of AΓ .
Theorem 3.3. There is an exact sequence
1 −→ PΠAΓ −→ CΓ (ι) −→ GL(n, Z/2).

(2)

Moreover, the image of CΓ (ι) in GL(n, Z/2) is generated by the images of all diagram
automorphisms and adjacent dominated transvections in Aut(AΓ ).
Proof. Let Φ2 : Aut(AΓ ) → GL(n, Z/2) be the map induced by the mod 2 abelianisation
map AΓ → (Z/2)n . We will show that PΠAΓ is the kernel of the restriction of Φ2 to CΓ (ι).
10

Let α ∈ CΓ (ι). Note that for each v ∈ V , the element α(v) necessarily has odd length,
since α(v) must survive under the mod 2 abelianisation map AΓ → (Z/2)n . Now for
each v ∈ V , write α(v) in clique-palindromic normal form w1 . . . wk . . . w1 , as in Proposition 3.1. Both the index k and the word wk here depend upon v, so we write w(v) for the
central clique word in the clique-palindromic normal form for α(v). Then each word w(v)
is a palindrome of odd length which is supported on a clique in Γ. It follows that the
automorphism α lies in PΠAΓ if and only if for each v ∈ V , the exponent sum of v in the
word w(v) is odd, and every other exponent sum is even. Thus PΠAΓ is precisely the kernel
of the restriction of Φ2 .
We now derive the generating set for Φ2 (CΓ (ι)) in the statement of the theorem. Given
α ∈ CΓ (ι), write α = δβ, where δ ∈ DΓ and β ∈ Aut0 (AΓ ). We map β into GL(n, Z/2) using
the canonical map Φ2 , and give Φ2 (β) the lower block-triangular decomposition discussed
in Section 2.4.
By Corollary 3.2, we can reduce each diagonal block of Φ2 (β) to an identity matrix by
composing Φ2 (β) with appropriate members of Φ2 (CΓ (ι)): permutation matrices (in the
case of a free block), or images of adjacent transvections (in the case of an abelian block).
The resulting matrix N ∈ Φ2 (CΓ (ι)) lifts to some α0 ∈ CΓ (ι).
If N has an off-diagonal 1 in its ith column, this corresponds to α0 (vi ) having odd exponent
sum of both vi and vj , say. Writing α0 (vi ) in clique-palindromic normal form w1 . . . wk . . . w1 ,
we must have that vi and vj both have odd exponent sum in wk , and hence commute, by
Proposition 3.1. The presence of the 1 in the (j, i) entry of N implies that vi ≤ vj , and so
we can use the image of the (adjacent) transvection τij to clear it.
Thus we conclude that Φ2 (β) may be written as a product of images of diagram automorphisms and adjacent transvections. Hence Φ2 (CΓ (ι)) is also generated by these automorphisms.
We now use Theorem 3.3 to prove that the collection of palindromic automorphisms ΠAΓ
is a subgroup of Aut(AΓ ). We will require the following result.
Proposition 3.4. Let α ∈ Aut(AΓ ) be palindromic. Then α can be expressed as α = δγ
where γ ∈ PΠAΓ and δ ∈ DΓ .
Proof. Let α ∈ ΠAΓ . Define a function δ : V → V by letting δ(v) be the middle letter
of a reduced palindromic word representing α(v). Note that δ is well-defined, because all
reduced expressions for α(v) are shuffle-equivalent, and in any such reduced expression there
is exactly one letter with odd exponent sum. The map δ must be bijective, otherwise the
image of α in GL(n, Z/2) would have two identical columns. We now show that δ induces
a diagram automorphism of AΓ , which by abuse of notation we also denote δ.
Since δ : V → V is a bijection and Γ is simplicial, it suffices to show that δ induces a
graph endomorphism of Γ. Suppose that u, v ∈ V are joined by an edge in Γ. Then
[α(v), α(u)] = 1, and so we apply Servatius’ Centraliser Theorem (Theorem 2.1). Write
α(u) in basic form w1 r1 . . . ws rs (see Section 2.2). Since α(u) is a palindrome, all but one
of these wi will be an even length palindrome, and exactly one will be an odd length
palindrome, with odd exponent sum of δ(u). We know by the Centraliser Theorem that

11

α(v) lies in
hw1 i × · · · × hws i × hlk(α(u))i.
Since δ(v) 6= δ(u), the only way α(v) can have an odd exponent of δ(v) is if δ(v) ∈ lk(α(u)).
In particular, [δ(v), δ(u)] = 1. Thus δ preserves adjacency in Γ and hence induces a diagram
automorphism.
The proposition now follows, setting γ = δ −1 α ∈ PΠAΓ .
The following corollary is immediate.
Corollary 3.5. The set ΠAΓ forms a group. Moreover, this group splits as PΠAΓ o DΓ .
We are now able to determine precisely when the groups ΠAΓ and CΓ (ι) appearing in the
exact sequence (2) in the statement of Theorem 3.3 are equal.
Proposition 3.6. The groups ΠAΓ and CΓ (ι) are equal if and only if Γ has no adjacent
domination classes.
Proof. If Γ has an adjacent domination class, then the adjacent transvections to which it
gives rise are in CΓ (ι) but not in ΠAΓ .
For the converse, suppose α ∈ CΓ (ι)\ΠAΓ . Write α = δβ, where δ ∈ DΓ and β ∈ Aut0 (AΓ ),
as in the proof of Theorem 3.3. Note that since DΓ ≤ CΓ (ι) we have that β ∈ CΓ (ι). There
must be a v ∈ V such that β(v) has at least two letters of odd exponent sum, say u1 and u2 ,
as otherwise α would lie in ΠAΓ . Recall that u1 and u2 must commute, as they both must
appear in the central clique word of the clique-palindromic normal form of β(v), in order
to have odd exponent.
Consider Φ(β) in GL(n, Z) under our usual lower block-triangular matrix decomposition,
discussed in Section 2.4. It must be the case that both u1 and u2 dominate v. This is
because the odd entries in the column of Φ(β) corresponding to v that arise due to u1
and u2 either lie in the diagonal block containing v, or below this block. In the former
case, this gives u1 , u2 ∈ [v], while in the latter, the presence of non-zero entries below the
diagonal block of v forces u1 , u2 ≥ v (as discussed in Section 2.4). If v dominates u1 , say,
in return, then we obtain u1 ≤ v ≤ u2 , and so by transitivity u1 is (adjacently) dominated
by u2 , proving the proposition in this case.
Now consider the case that neither u1 nor u2 is dominated by v. By Corollary 3.2, we may
carry out some sequence of row operations to Φ(β) corresponding to the images of inversions,
adjacent transvections, or Pij in Φ(CΓ (ι)), to reduce the diagonal block corresponding to [v]
to the identity matrix. The resulting matrix lifts to some β 0 ∈ CΓ (ι), such that β 0 (v) has
exponent sum 1 of v, and odd exponent sums of u1 and of u2 . As we argued in the proof of
Corollary 3.2, this means u1 , u2 and v pairwise commute, and so v is adjacently dominated
by u1 (and u2 ). This completes the proof.

3.3

Finite generating sets

In this section we prove Theorem A of the introduction, which gives a finite generating set
for the palindromic automorphism group ΠAΓ . The main step is Theorem 3.7, where we
12

determine a finite set of generators for the pure palindromic automorphism group PΠAΓ .
We also obtain finite generating sets for the centraliser CΓ (ι) in Corollary 3.8, and for
certain stabiliser subgroups of ΠAΓ in Theorem 3.11.
Theorem 3.7. The group PΠAΓ is generated by the finite set comprising the inversions
and the dominated elementary palindromic automorphisms.
Before proving Theorem 3.7, we state a corollary obtained by combining Theorems 3.3
and 3.7.
Corollary 3.8. The group CΓ (ι) is generated by diagram automorphisms, adjacent dominated transvections and the generators of PΠAΓ .
Our proof of Theorem 3.7 is an adaptation of Laurence’s proof [16] of finite generation
of Aut(AΓ ). First, in Lemma 3.9 below, we show that any α ∈ PΠAΓ may be precomposed with suitable products of our proposed generators to yield what we refer to as a
‘simple’ automorphism of AΓ (defined below). The simple palindromic automorphisms may
then be understood by considering subgroups of PΠAΓ that fix certain free product subgroups inside AΓ ; we define and obtain generating sets for these subgroups in Lemma 3.10.
Combining these results, we complete our proof of Theorem 3.7.
For each v ∈ V , we define α ∈ PΠAΓ to be v-simple if supp(α(v)) is connected in Γc . We
say that α ∈ PΠAΓ is simple if α is v-simple for all v ∈ V . Laurence’s definition of a
v-simple automorphism φ ∈ Aut(AΓ ) is more general and differs from ours, however the
two definitions are equivalent when φ ∈ PΠAΓ .
Let S denote the set of inversions and dominated elementary palindromic automorphisms
in ΠAΓ (that is, the generating set for PΠAΓ proposed by Theorem 3.7). We say that
α, β ∈ PΠAΓ are π-equivalent if there exists θ ∈ hSi such that α = βθ. In other words,
α, β ∈ PΠAΓ are π-equivalent if β −1 α ∈ hSi.
Lemma 3.9. Every α ∈ PΠAΓ is π-equivalent to some simple automorphism χ ∈ PΠAΓ .
Proof. Suppose α ∈ PΠAΓ . We note once and for all that the palindromic word α(u) is
cyclically reduced, for any u ∈ V .
Select a vertex v ∈ V of maximal rank for which α(v) is not v-simple. Now write
α(v) = w1 r1 . . . ws rs
in basic form, reindexing if necessary so that v ∈ supp(w1 ). The ranks of v and α(v) are
equal, since α induces an isomorphism from the centraliser in AΓ of v to that of α(v). Hence
by Proposition 2.2, parts 2(b) and 2(a) respectively, each wi ∈ AΓ (for i > 1) is some vertex
generator in V , and wi ≥0 v. Moreover, for i > 1, each ri is even, since α(v) is palindromic.
Now, for i > 1, suppose wi ≥0 v but [v]0 6= [wi ]0 . By Servatius’ Centraliser Theorem
(Theorem 2.1), we know that the centraliser of a vertex is generated by its star, and hence
conclude that rk(wi ) > rk(v). This gives that α is wi -simple, by our assumption on the
maximality of the rank of v. In basic form, then,
α(wi ) = p` ,
13

where ` ∈ Z, p ∈ AΓ , and supp(p) is connected in Γc . Note also that supp(p) contains wi ,
since α ∈ PΠAΓ .
Suppose there exists t ∈ supp(p) \ {wi }. As for v before, by Proposition 2.2, we have t ≥ wi ,
since rk(α(wi )) = rk(wi ). We know wi ≥0 v, and so t ≥ v. Since wi , v and t are pairwise
distinct, this forces wi and t to be adjacent, which contradicts Proposition 2.2, part 2(c).
So
α(wi ) = wi ` ,
and necessarily ` = ±1. Knowing this, we replace α with αβi where βi ∈ hSi is the
palindromic automorphism of the form
v 7→ wi

`ri
2

vwi

`ri
2

.

By doing this for each such wi , we ensure that any wi that strictly dominates v is not in
the support of αβi (v). Note α(v 0 ) = αβi (v 0 ) for all v 0 6= v.
If s = 1, then α is v-simple, so by our assumption on v, we must have s > 1. Because
we have reduced to the case where wi ∈ [v]0 for i > 1, we must have w1 = v ±1 , otherwise
we get a similar adjacency contradiction as in the previous paragraph: if there exists t ∈
supp(w1 ) \ {v}, then, as before, t ≥ v, and since [wi ]0 = [v]0 , this would force t and v to
be adjacent. Thus α(v) ∈ h[v]0 i. Indeed, the discussion in the previous two paragraphs
goes through for any u ∈ [v]0 , so we may assume that α(u) ∈ h[v]0 i for any u ∈ [v]0 . Thus
αh[v]0 i ≤ h[v]0 i, with equality holding by [16, Proposition 6.1].
The group h[v]0 i is free abelian, and by considering exponent sums, we see that the restriction
of α to the group h[v]0 i is a member of the level 2 congruence subgroup Λk [2], where k = |[v]0 |.
We know that Theorem 3.7 holds in the special case of these congruence groups (see [9,
Lemma 2.4], for example), so we can precompose α with the appropriate automorphisms in
the set S so that the new automorphism obtained, α0 , is the identity on h[v]0 i, and acts the
same as α on all other vertices in V . The automorphisms α and α0 are π-equivalent, and α0
is v-simple (indeed: α0 (v) = v).
From here, we iterate this procedure, selecting a vertex u ∈ V \ {v} of maximal rank for
which α0 is not u-simple, and so on, until we have exhausted the vertices of Γ preventing α
from being simple.
Now, for each v ∈ V , define Γv be the set of vertices that dominate v but are not adjacent
to v. Further define Xv := {v = v1 , . . . , vr } ⊆ Γv to be the vertices of Γv that are also
dominated by v. Partition Γv into its connected components in the graph Γ \ lk(v). This
partition is of the form
!
!
t
r
G
G
Γi t
{vi } ,
i=1

where

Ft

i=1 Γi

i=1

= Γv \ Xv . Letting Hi = hΓi i, we see that
H := hΓv i = H1 ∗ · · · ∗ Ht ∗ hXv i,

(3)

where Fr := hXv i is a free group of rank r. Notice that H is itself a right-angled Artin
group.
14

The final step in proving Theorem 3.7 requires a generating set for a certain subgroup of
palindromic automorphisms in Aut(H), which we now define. Let Y denote the subgroup of
Aut(H) consisting of the pure palindromic automorphisms of H that restrict to the identity
on each Hi . The following lemma says that this group is generated by its intersection with
the finite list of generators stated in Theorem 3.7. In the special case when there are no Hi
factors in the free product (3) above, this result was established by Collins [5]. Our proof
is a generalisation of his.
Lemma 3.10. The group Y is generated by the inversions of the free group Fr and the
elementary palindromic automorphisms of the form P (s, t) : s 7→ tst, where t ∈ Γv and
s ∈ Xv .
Proof. For α ∈ Y, we define its length l(α) to be the sum of the lengths of α(vi ) for each
vi ∈ Xv . We induct on this length. The base case is l(α) = r, in which case α is a product
of inversions of Fr . From now on, assume l(α) > r.
Let L(w) denote the length of a word w in the right-angled Artin group H, with respect to
the vertex set Γv . Suppose for all i , j ∈ {±1} and distinct ai , aj ∈ α(Γv ) we have


L(ai i ajj ) > L(ai ) + L(aj ) − 2(bL(ai )/2c + 1),

(4)

where bxc is the integer part of x ∈ [0, ∞). Conceptually, we are assuming that for every


expression ai i ajj , whatever cancellation occurs between the words ai i and ajj , more than

half of ai i and more than half of ajj survives after all cancellation is complete.
Fix vi ∈ Xv so that ai := α(vi ) satisfies L(ai ) > 1. Such a vertex vi must exist, as we are
assuming that l(α) > r. Notice that since L(ai ) > 1, we have vi 6= a±1
i . Now, any reduced
word in H of length m with respect to the generating set α(Γv ) has length at least m with
respect to the vertex generators Γv , due to our cancellation assumption. Since vi 6= ai ±1 ,
the generator vi must have length strictly greater than 1 with respect to α(Γv ), and so vi
must have length strictly greater than 1 with respect to Γv . But vi is an element of Γv ,
which is a contradiction. Therefore, the above inequality (4) fails at least once.
We now argue each case separately. Let ai , aj ∈ α(Γv ) be distinct and write
ai = α(vi ) = wi vi ηi wi rev

and aj = α(vj ) = wj vj ηj wj rev ,

where vi , vj ∈ Γv , wi , wj ∈ H and ηi , ηj ∈ {±1}. Suppose the inequality (4) fails for this
pair when i = j = 1. Then it must be the case that wj = (wi rev )−1 vi−ηi z, for some z ∈ H,
since H is a free product. In this case, replacing α with αP (vj , vi ) = αPji decreases the
length of the automorphism. We reduce the length of α in the remaining cases as follows:
• For i = j = −1, replace α with αιj P (vj , vi )−1 = αιj Pji−1 .
• For i = −1 and j = 1, or vice versa, replace α with αιj P (vj , vi ) = αιj Pji .
By induction, we have thus established the proposed generating set for the group Y.
We now prove Theorem 3.7, obtaining a finite generating set for the group PΠAΓ .
15

Proof of Theorem 3.7. Let S denote the set of inversions and dominated elementary palindromic automorphisms in PΠAΓ . By Lemma 3.9, all we need do is write any simple
α ∈ PΠAΓ as a product of members of S ±1 .
Let v be a vertex of maximal rank that is not fixed by α. Define Γv , its partition, and the
free product it generates using the same notation as in the discussion before the statement
of Lemma 3.10. By maximality of the rank of v, any vertex of any Γi must be fixed by α
(since it has rank higher than that of v). By Lemma 5.5 of Laurence and its corollary [16],
we conclude that (for this v we have chosen), α(H) = H.
This establishes that α restricted to H lies in the group Y ≤ Aut(H), for which Lemma 3.10
gives a generating set. Thus we are able to precompose α with the appropriate members
of S ±1 to obtain a new automorphism α0 that is the identity on H, and which agrees with α
on Γ \ Γv . In particular, α0 fixes v. We now iterate this procedure until all vertices of Γ are
fixed, and have thus proved the theorem.
With Theorem 3.7 established, we are now able to prove our first main result, Theorem A,
and so obtain our finite generating set for ΠAΓ .
Proof of Theorem A. By Corollary 3.5, we have that ΠAΓ splits as
ΠAΓ ∼
= PΠAΓ o DΓ ,
and so to generate ΠAΓ , it suffices to combine the generating set for PΠAΓ given by Theorem 3.7 with the diagram automorphisms of AΓ . Thus the group ΠAΓ is generated by
the set of all diagram automorphisms, inversions and well-defined dominated elementary
palindromic automorphisms.
We end this section by remarking that the proof techniques we used in establishing Theorem A allow us to obtain finite generating sets for a more general class of palindromic
automorphism groups of AΓ . Having chosen an indexing v1 , . . . , vn of the vertex set V of Γ,
denote by ΠAΓ (k) the subgroup of ΠAΓ that fixes each of the vertices v1 , . . . , vk . Note that
a reindexing of V will, in general, produce non-isomorphic stabiliser groups. We are able
to show that each ΠAΓ (k) is generated by its intersection with the finite set S.
Theorem 3.11. The stabiliser subgroup ΠAΓ (k) is generated by the set of diagram automorphisms, inversions and dominated elementary palindromic automorphisms that fix each
of v1 , . . . , vk .
Throughout the proof of Theorem 3.7, each time that we precomposed some α ∈ PΠAΓ
by an inversion ιi , an elementary palindromic automorphism Pij , or its inverse Pij−1 , it was
because the generator vi was not fixed by α. If vj ∈ V was already fixed by α, we had no
±1
need to use ιj or any of the Pjk
(j 6= k) in this way. (That this claim holds in the secondlast paragraph of the proof of Lemma 3.9, where we are working in the group Λk [2], follows
from [9, Lemma 3.5].) The same is true when we extend PΠAΓ to ΠAΓ using diagram
automorphisms, in the proof of Theorem A. Thus by following the same method as in our
proof of Theorem A, we are also able to obtain the more general result, Theorem 3.11: our
approach had already written α ∈ ΠAΓ (k) as a product of the generators proposed in the
statement of Theorem 3.11.
16

4

The palindromic Torelli group

Recall that we defined the palindromic Torelli group PI Γ to consist of the palindromic
automorphisms of AΓ that act trivially on H1 (AΓ , Z). Our main goal in this section is to
prove Theorem B, which gives a generating set for PI Γ . For this, in Section 4.1 we obtain a
finite presentation for the image in GL(n, Z) of the pure palindromic automorphism group.
Using the relators from this presentation, we then prove Theorem B in Section 4.2.

4.1

Presenting the image in GL(n, Z) of the pure palindromic automorphism group

In this section we prove Theorem 4.2, which establishes a finite presentation for the image
of the pure palindromic automorphism group PΠAΓ in GL(n, Z), under the canonical map
induced by abelianising AΓ . Corollary 4.3 then gives a splitting of PΠAΓ .
Recall that Λn [2] denotes the principal level 2 congruence subgroup of GL(n, Z). We start
by recalling a finite presentation for Λn [2] due to the first author. For 1 ≤ i 6= j ≤ n, let
Sij ∈ Λn [2] be the matrix that has 1s on the diagonal and 2 in the (i, j) position, with 0s
elsewhere, and let Zi ∈ Λn [2] differ from the identity matrix only in having −1 in the (i, i)
position. Theorem 4.1 gives a finite presentation for Λn [2] in terms of these matrices.
Theorem 4.1 (Fullarton [9]). The principal level 2 congruence group Λn [2] is generated by
{Sij , Zi | 1 ≤ i 6= j ≤ n},
subject to the defining relators
1. Zi 2

6. [Ski , Skj ]

2. [Zi , Zj ]

7. [Sij , Skl ]

3. (Zi Sij )2

8. [Sji , Ski ]

4. (Zj Sij )2

9. [Skj , Sji ]Ski −2
10. (Sij Sik −1 Ski Sji Sjk Skj −1 )2

5. [Zi , Sjk ]
where 1 ≤ i, j, k, l ≤ n are pairwise distinct.

We will use this presentation of Λn [2] to obtain a finite presentation of the image of PΠAΓ
in GL(n, Z). Observe that ιj 7→ Zj and Pij 7→ Sji (vi ≤ vj ) under the canonical map
Φ : Aut(AΓ ) → GL(n, Z). Let RΓ be the set of words obtained by taking all the relators in
Theorem 4.1 and removing those that include a letter Sji with vi 6≤ vj .
Theorem 4.2. The image of PΠAΓ in GL(n, Z) is a subgroup of Λn [2], with finite presentation
h{Zk , Sji : 1 ≤ k ≤ n, vi ≤ vj } | RΓ i .

17

Proof. By Theorem 3.7, we know that PΠAΓ ≤ Aut0 (AΓ ), and so matrices in Θ :=
Φ(PΠAΓ ) ≤ GL(n, Z) may be written in the lower-triangular block decomposition discussed in Section 2.4. Moreover, the matrix in a diagonal block of rank k in some A ∈ Θ
must lie in Λk [2].
We now use this block decomposition to obtain the presentation of Θ in the statement of
the theorem. Observe that we have a forgetful map F defined on Θ, where we forget the
first k := |[v1 ]| rows and columns of each matrix. This is a well-defined homomorphism,
since the determinant of a lower block-triangular matrix is the product of the determinants
of its diagonal blocks. Let Q denote the image of this forgetful map, and K its kernel. We
have K = Λk [2] × Zt , where t is the number of dominated transvections that are forgotten
under the map F, and the Λk [2] factor is generated by the images of the inversions and
dominated elementary palindromic automorphisms that preserve the subgroup h[v1 ]i.
The group Θ splits as K o Q, with the relations corresponding to the semi-direct product
action, and those in the obvious presentation of K, all lying in RΓ . Now, we may define a
similar forgetful map on the matrix group Q, so by induction Λ is an iterated semi-direct
product, with a complete set of relations given by RΓ .
Using the above presentation, we are able to obtain the following corollary, regarding a
splitting of the group PΠAΓ . Recall that IΓ is the subgroup of Aut(AΓ ) generated by inversions. We denote by EΠAΓ the subgroup of PΠAΓ generated by all dominated elementary
palindromic automorphisms.
Corollary 4.3. The group PΠAΓ splits as EΠAΓ o IΓ .
Proof. The group PΠAΓ is generated by EΠAΓ and IΓ by Theorem 3.7, and IΓ normalises EΠAΓ . We now establish that EΠAΓ ∩ IΓ is trivial. Suppose α ∈ EΠAΓ ∩ IΓ .
By Theorem 4.2, the image of α under the canonical map Φ : Aut(AΓ ) → GL(n, Z) lies in
the principal level 2 congruence group Λn [2]. This implies that Φ(α) is trivial, since Λn [2]
is itself a semi-direct product of groups containing the images of the groups EΠAΓ and IΓ ,
respectively: this is verified by examining the presentation of Λn [2] given in Theorem 4.1.
So the automorphism α must lie in the palindromic Torelli group PI Γ , which has trivial
intersection with IΓ , and hence α is trivial.

4.2

A generating set for the palindromic Torelli group

Using the relators in the presentation given by Theorem 4.1, we are now able to obtain an
explicit generating set for the palindromic Torelli group PI Γ , and so prove Theorem B.
Recall that when AΓ is a free group, the elementary palindromic automorphism Pij is welldefined for every distinct i and j. The first author defined doubled commutator transvections
and separating π-twists in Aut(Fn ) (n ≥ 3) to be conjugates in ΠAn of, respectively, the automorphisms [P12 , P13 ] and (P23 P13 −1 P31 P32 P12 P21 −1 )2 . The latter of these two may seem
cumbersome; we refer to [9, Section 2] for a simple, geometric interpretation of separating
π-twists.
The definitions of these generators extend easily to the general right-angled Artin groups
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setting, as follows. Suppose vi ∈ V is dominated by vj and by vk , for distinct i, j and k.
Then
χ1 (i, j, k) := [Pij , Pik ] ∈ Aut(AΓ )
is well-defined, and we define a doubled commutator transvection in Aut(AΓ ) to be a conjugate in ΠAΓ of any well-defined χ1 (i, j, k). Similarly, suppose [vi ] = [vj ] = [vk ] for distinct
i, j and k. Then
χ2 (i, j, k) := (Pjk Pik −1 Pki Pkj Pij Pji −1 )2 ∈ Aut(AΓ )
is well-defined, and we define a separating π-twist in Aut(AΓ ) to be a conjugate in ΠAΓ of
any well-defined χ2 (i, j, k).
We now prove Theorem B, showing that PI Γ is generated by these two types of automorphisms.
Proof of Theorem B. Recall that Θ := Φ(PΠAΓ ) ≤ GL(n, Z). The images in Θ of our
generating set for PΠAΓ (Theorem 3.7) form the generators in the presentation for Θ
given in Theorem 4.2. Thus using a standard argument (see, for example, the proof of
[17, Theorem 2.1]), we are able to take the obvious lifts of the relators of Θ as a normal
generating set of PI Γ in PΠAΓ , via the short exact sequence
1 −→ PI Γ −→ PΠAΓ −→ Θ −→ 1.
The only such lifts and their conjugates that are not trivial in PΠAΓ are the ones of the
form stated in the theorem.

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