arXiv:1611.01038v1 [] 3 Nov 2016

CLASSIFICATION OF CURTIS-TITS AND PHAN
AMALGAMS WITH 3-SPHERICAL DIAGRAM
RIEUWERT J. BLOK, CORNELIU G. HOFFMAN, AND SERGEY V. SHPECTOROV
Abstract. We classify all non-collapsing Curtis-Tits and Phan amalgams
with 3-spherical diagram over all fields. In particular, we show that amalgams with spherical diagram are unique, a result required by the classification of finite simple groups. We give a simple condition on the amalgam
which is necessary and sufficient for it to arise from a group of Kac-Moody
type. This also yields a definition of a large class of groups of Kac-Moody
type in terms of a finite presentation.

1. Introduction
Local recognition results play an important role in various parts of mathematics. A key example comes from the monumental classiﬁcation of ﬁnite simple
groups. Local analysis of the unknown ﬁnite simple group G yields a local
datum consisting of a small collection of subgroups ﬁtting together in a particular way, called an amalgam. The Curtis-Tits theorem [7, 16, 17, 18, 19]
and the Phan (-type) theorems [25, 26, 27] describe amalgams appearing in
known groups of Lie type. Once the amalgam in G is identiﬁed as one of the
amalgams given by these theorems, G is known.
The present paper was partly motivated by a question posed by R. Solomon
and R. Lyons about this identiﬁcation step, arising from their work on the
classiﬁcation [9, 10, 11, 12, 13, 14]: Are Curtis-Tits and Phan type amalgams
uniquely determined by their subgroups? More precisely is there a way of
ﬁtting these subgroups together so that the amalgam gives rise to a diﬀerent
group? In many cases it is known that, indeed, depending on how one ﬁts the
subgroups together, either the resulting amalgam arises from these theorems,
or it does not occur in any non-trivial group. This is due to various results
of Bennet and Shpectorov [1], Gramlich [15], Dunlap [8], and R. Gramlich,
M. Horn, and W. Nickel [23]. However, all of these results use, in essence, a
crucial observation by Bennett and Shpectorov about tori in rank-3 groups of
Lie type, which fails to hold for small ﬁelds. In the present paper we replace
the condition on tori by a more eﬀective condition on root subgroups, which
holds for all ﬁelds. This condition is obtained by a careful analysis of maximal
subgroups of groups of Lie type. Thus the identiﬁcation step can now be made
for all possible ﬁelds. A useful consequence of the identiﬁcation of the group
1

2

BLOK, HOFFMAN, AND SHPECTOROV

G, together with the Curtis-Tits and Phan type theorems, is that it yields a
simpliﬁed version of the Steinberg presentation for G.
Note that this solves the - generally much harder - existence problem: “how
can we tell if a given amalgam appears in any non-trivial group?”
The uniﬁed approach in the present paper not only extends the various
results on Curtis-Tits and Phan amalgams occurring in groups of Lie type
to arbitrary ﬁelds, but in fact also applies to a much larger class of CurtisTits and Phan type amalgams, similar to those occurring in groups of KacMoody type. Here, both the uniqueness and the existence problem become
signiﬁcantly more involved.
Groups of Kac-Moody type were introduced by J. Tits as automorphism
groups of certain geometric objects called twin-buildings [29]. In the same
paper J. Tits conjectured that these groups are uniquely determined by the
fact that the group acts on some twin-building, together with local geometric
data called Moufang foundations. As an example he sketched an approach
towards classifying such foundations in the case of simply-laced diagrams. This
conjecture was subsequently proved for Moufang foundations built from locally
split and locally ﬁnite rank-2 residues by B. Mühlherr in [24] and reﬁned by
P. E. Caprace in [6]. All these results produce a classiﬁcation of groups of
Kac-Moody type using local data in the form of an amalgam, together with
a global geometric assumption stipulating the existence of a twin-building on
which the group acts.
Ideally, one would use the generalizations of the Curtis-Tits and Phan type
theorems to describe the groups of Kac-Moody type in terms of a simpliﬁed
Steinberg type presentation. However, the geometric assumption is unsatisfactory for this purpose as it is impossible to verify directly from the presentation
itself.
In our uniﬁed approach we consider all possible amalgams whose local structure is any one of those appearing in the above problems. There is no condition
on the ﬁeld. Then, we classify those amalgams that satisfy our condition on
root groups and show that in the spherical case they are unique. This explains
why groups of Lie type can uniquely be recognized by their amalgam. By
contrast, in the non-spherical case the amalgams are not necessarily unique
and, indeed, not all such amalgams give rise to groups of Kac-Moody type.
This is a consequence of the fact that we impose no global geometric condition. Nevertheless, we give a simple condition on the amalgam itself which
decides whether it comes from a group of Kac-Moody type or not. As a result, we obtain a purely group theoretic deﬁnition of a large class of groups of
Kac-Moody type just in terms of a ﬁnite presentation.
Finally, we note that an amalgam must satisfy the root subgroup condition
to occur in a non-trivial group. A subsequent study generalizing [3, 5] shows

CURTIS-TITS AND PHAN AMALGAMS

3

that in fact all amalgams satisfying the root group condition do occur in nontrivial groups. Thus, in this much wider context the existence problem is also
solved.
We shall now give an overview of the results in the present paper. Recall
that a Dynkin diagram Γ is an oriented edge-labelled graph. We say that Γ is
connected if the underlying (unlabelled) graph is connected in the usual sense.
Moreover, we use topological notions such as spanning tree and homotopy rank
of Γ referring to the underlying graph.
For Phan amalgams we prove the following (for the precise statement see
Theorem 5.21).
Theorem A. Let q be any prime power and let Γ be a connected 3-spherical
diagram
with homotopy rank r. Then, there is a bijection between the elements
Q
of rs=1 Aut(Fq2 ) and the type preserving isomorphism classes of Curtis-Tits
amalgams with diagram Γ over Fq .
For Curtis-Tits amalgams the situation is slightly more complicated (for the
precise statement see Theorem 4.24).
Theorem B. Let q be a prime power and let Γ be a connected 3-spherical
diagram with homotopy rank r. Then there exists a set of Q
positive integers
{e1 , . . . , er } so that there is a bijection between the elements of rs=1 Aut(Fqes )×
Z/2Z and the type preserving isomorphism classes of Curtis-Tits amalgams
with diagram Γ over Fq .
Corollary C. Let q be a prime power and let Γ be a 3-spherical tree. Then,
up to type preserving isomorphism, there is a unique Curtis-Tits and a unique
Phan amalgam over Fq with diagram Γ.
Note that Corollary C includes all spherical diagrams of rank ≥ 3. Several
special cases of the above results were proved elsewhere. Indeed, Theorem B
was proved for simply-laced diagrams and q ≥ 4 in [4]. Corollary C was proved
for Phan amalgams with Γ = An in [1], for general simply-laced tree diagram
in [8], and for Γ = Cn for q ≥ 3 in [20, 22].
The classiﬁcation of Curtis-Tits amalgams will be done along the following
lines. Note if (G, Gi , Gj ) is a Curtis-Tits standard of type diﬀerent from
A1 × A1 , and X is any Sylow p-subgroup in one of the vertex groups, say
Gi , then generically it generates G together with Gj . In Subsection 4.1 we
+
−
show that there is a unique pair (Xi , Xi ) of Sylow p-subgroups in Gi whose
members do not have this property. Moreover, each member commutes with
a unique member in the other vertex group.
In Subsection 4.2 we show that in a non-collapsing Curtis-Tits amalgam
−
G = {Gi , Gi,j , gi,j | i, j ∈ I} for each i there exists a pair (X+
i , Xi ) of Sylow

4

BLOK, HOFFMAN, AND SHPECTOROV

−
subgroups in Gi such that for any edge {i, j} (gi,j (X+
i , Xi ) is the pair for
+
−
(Gi,j , Gi , Gj ) as above. The collection X = {Xi , Xi : i ∈ I} is called a weak
system of fundamental root groups. Without loss of generality one can assume
that any amalgam with the same diagram has the exact same weak system
X . As a consequence all amalgams with the same diagram can be determined
up to isomorphism by studying the coeﬃcient system associated to X , that
is, the graph of groups consisting of automorphisms of the vertex and edge
groups preserving X . In Subsection 4.3 we determine the coeﬃcient system
associated to X . In Subsection 4.4, we pick a spanning tree Σ for Γ and use
precise information about the coeﬃcient system to create a standard form
of a Curtis-Tits amalgam in which all vertex-edge inclusion maps are trivial
except for the edges in Σ. In particular this shows that if Γ is a tree, then the
amalgam is unique up to isomorphism. Finally in Subsection 4.5 we show that
for a suitable choice of Σ, the remaining non-trivial inclusion maps uniquely
determine the amalgam.
The classiﬁcation of Phan amalgams in Section 5 follows the same pattern.
However in this case the role of the weak system of fundamental root groups
is replaced by a system of tori in the vertex groups, whose images in the edge
groups must form a torus there.

As shown here, the existence of a weak system of fundamental root groups is
a necessary condition for the existence of a non-trivial completion. A natural
question of course is whether it is also suﬃcient. In the spherical cases, the
amalgams are unique and the Curtis-Tits and Phan theorems identify universal completions of these amalgams. In [5] it is shown that any Curtis-Tits
amalgam with 3-spherical simply-laced diagram over a ﬁeld with at least four
elements having property (D) has a non-trivial universal completion, which is
identiﬁed up to a rather precisely described central extension. In the present
paper we will not study completions of the Curtis-Tits and Phan amalgams
classiﬁed here, but merely note that similar arguments yield non-trivial completions for all amalgams. In particular, the conditions mentioned above are
indeed suﬃcient for the existence of these completions. In general we don’t
know of a direct way of giving conditions on an amalgam ensuring the existence
of a non-trivial completion.

Acknowledgements
This paper was written as part of the project KaMCAM funded by the European Research Agency through a Horizon 2020 Marie-Sklodowska Curie fellowship (proposal number 661035).

CURTIS-TITS AND PHAN AMALGAMS

5

2. Curtis-Tits and Phan amalgams and their diagrams
2.1. Diagrams
In order to ﬁx some notation, we start with some deﬁnitions.
Definition 2.1. A Coxeter matrix over the set I = {1, 2, . . . , n} of ﬁnite
cardinality n is a symmetric matrix M = (mij )i,j∈I with entries in N≥1 ∪ {∞}
such that, for all i, j ∈ I distinct we have mii = 1 and mij ≥ 2.
A Coxeter diagram with Coxeter matrix M is an edge-labelled graph ∆ =
(I, E) with vertex set I = V ∆ and edge-set E = E ∆ without loops such that
for any distinct i, j ∈ I, there is an edge labelled mij between i and j whenever
mij > 2; if mi,j = 2, there is no such edge. Thus, M and ∆ determine each
other uniquely. For any subset J ⊆ I, we let ∆J denote the diagram induced
on vertex set J. We say that ∆ is connected if the underlying (unlabelled)
graph is connected in the usual sense. Moreover, we use topological notions
such as spanning tree and homotopy rank of ∆ referring to the underlying
graph.
A Coxeter system with Coxeter matrix M is a pair (W, S), where W is a
group generated by the set S = {si : i ∈ I} subject to the relations (si sj )mij =
1 for all i, j ∈ I. For each subset J ⊆ I, we let WJ = hsj : j ∈ JiW . We
call M and (W, S) m-spherical if every subgroup WJ with |J| = m is ﬁnite
(m ∈ N≥2 ). Call (W, S) spherical if it is n-spherical.
In order to describe Curtis-Tits and Phan amalgams more precisely, we also
introduce a Lie diagram.
Definition 2.2. Let ∆ = (I, E) be a Coxeter diagram. A Lie diagram of
Coxeter type ∆ is an untwisted or twisted Dynkin diagram Γ whose edge labels
lij do not specify the orientation. In this paper we shall only be concerned
with Lie diagrams of Coxeter type An , Bn , Dn , E6 , E7 , E8 , and F4 . For these,
we have the following correspondence
∆
Γ

An
An

Bn
Bn , Cn , 2Dn+1 , 2A2n−1 , 2A2n

Dn , En (n = 6, 7, 8)
F4
Dn
En
F4 , F4∗ , 2E6 , 2E6∗

Here F4 and 2E6 (resp. F4∗ and 2E6∗ ) denote the diagrams where node 1 corresponds to the long (resp. short) root (Bourbaki labeling).
Let us introduce some more notation. We shall denote the Frobenius automorphism of order 2 of Fq2 by σ. Below we will consider sesquilinear forms h
on an Fq2 -vector space V . By convention, all these forms are linear in the ﬁrst
coordinate, that is h(λu, µv) = λh(u, v)µσ for u, v ∈ V and λ, µ ∈ Fq2 . Recall
that h is hermitian if h(v, u) = h(u, v)σ for all u, v ∈ V .

6

BLOK, HOFFMAN, AND SHPECTOROV

2.2. Standard pairs of Curtis-Tits type
Let Γ be a Lie diagram of type A2 , B2 /C2 , 2D3 /2A3 and q = pe for some prime
p ∈ Z and e ∈ Z≥1 . Then a Curtis-Tits standard pair of type Γ(q) is a triple
(G, G1 , G2 ) of groups such that one of the following occurs:
(Γ = A1 × A1 ). Now G = G1 × G2 and G1 ∼
= SL2 (q).
= G2 ∼
(Γ = A2 ). Now G = SL3 (q) = SL(V ) for some Fq -vector space with basis
{e1 , e2 , e3 } and G1 (resp. G2 ) is the stabilizer of the subspace he1 , e2 i (resp.
he2 , e3 i) and the vector e3 (resp. e1 ).
Explicitly we have



 a b



c d
G1 =
: a, b, c, d ∈ Fq with ad − bc = 1 ,


1



 1



a b : a, b, c, d ∈ Fq with ad − bc = 1 .
G2 =


c d
(Γ = C2 ). Now G = Sp4 (q) = Sp(V, β), where V is an Fq -vector space with
basis {e1 , e2 , e3 , e4 } and β is the symplectic form with Gram matrix


1 0

0 1
.
M =

−1 0
0 −1

G1 ∼
= SL2 (q) is the derived subgroup of StabG (he1 , e2 i) ∩ StabG (he3 , e4 i) and
G2 = StabG (e1 ) ∩ StabG (e3 ) ∼
= SL2 (q). Explicitly we have
= Sp2 (q) ∼



a b







c
d


: a, b, c, d ∈ Fq with ad − bc = 1 ,
G1 = 
d −c






−b a



1







a
b


.
:
a,
b,
c,
d
∈
F
with
ad
−
bc
=
1
G2 = 
q
1 






c
d

Remark 2.3. We are only interested in Curtis-Tits standard pairs of type
B2 for q odd. However, in that case we have Spin5 (q) ∼
= Sp4 (q) is the unique
PSp
(q).
Therefore, we can also
central extension of the simple group Ω5 (q) ∼
=
4
describe the Curtis-Tits standard pair for B2 as a Curtis-Tits standard pair
for C2 with G1 and G2 interchanged.

CURTIS-TITS AND PHAN AMALGAMS

7

(Γ = 2A3 ). Now G = SU4 (q) = SU(V ) for some Fq2 -vector space V with basis
{e1 , e2 , e3 , e4 } equipped with a non-degenerate hermitian form h for which this
basis is hyperbolic with Gram matrix

1 0

0 1
.
M =

1 0
0 1


Now G1 is the derived subgroup of the simultaneous stabilizer of the subspaces
he1 , e2 i and he3 , e4 i and G2 is the stabilizer of the vectors e1 and e3 and the
hyperbolic line he2 , e4 i. We have G2 ∼
= SL2 (q 2 ).
= SL2 (q) and G1 ∼
= SU2 (q) ∼
Explicitly we have



a b







c
d


G1 = 
: a, b, c, d ∈ Fq2 with ad − bc = 1 ,
dσ −cσ 






−bσ aσ



1







a
bη


:
a,
b,
c,
d
∈
F
with
ad
−
bc
=
1
,
G2 = 
q

1






cη −1
d
where η ∈ Fq2 has η + η q = 0.
Remark 2.4. For completeness we also deﬁne a standard Curtis-Tits pair
−
(H, H1, H2 ) of type 2D3 (q). Take H = Ω−
(V, Q), where V is an Fq 6 (q) = Ω P
vector space with basis {e1 , e2 , e3 , e4 , e5 , e6 } and Q( 5i=1 xi ei ) = x1 x3 +x2 x4 +
f (x5 , x6 ), for some quadratic polynomial f (x, 1) that is irreducible over Fq .
Here, H1 ∼
= SL2 (q) is the derived subgroup of StabH (he1 , e2 i) ∩ StabH (he3 , e4 i)
if SL2 (q) is perfect that is q > 2, and it is the subgroup StabH (he1 , e2 i) ∩
StabH (he3 , e4 i) ∩ Stab(v) for some non-singular vector v ∈ he5 , e6 i if q = 2,
2
∼
and H2 = StabH (e1 ) ∩ StabH (e3 ) ∼
= Ω−
4 (q) = PSL2 (q ).
However, there exists a unique standard Curtis-Tits pair (G, G1 , G2 ) of
type 2A3 (q) and a surjective homomorphism π : G → H with ker π = {±1} =
2
∼
Z(G1 ) ≤ Z(G). It induces π : G1 ∼
= SL2 (q 2 ) → Ω−
4 (q) = H2 = PSL2 (q ) and
∼
π : G2 = SU2 (q) → SL2 (q) = H2 . Because of this map, any amalgam involving
a standard Curtis-Tits pair of type 2D3 (q) is the image of an amalgam involing
a standard Curtis-Tits pair of type 2A3 (q) (see Subsection 2.4).

8

BLOK, HOFFMAN, AND SHPECTOROV

Definition 2.5. For Curtis-Tits amalgams, the standard identification map
will be the isomorphism g : SL2 (q e ) → Gi sending


a b
c d
to the corresponding matrix of Gi as described above. Here e = 1 unless
Γ(q) = 2A3 (q) and i = 1 or Γ(q) = 2D3 and i = 2, in which case e = 2.
2.3. Standard pairs of Phan type
Let Γ be as above. Then a Phan standard pair of type Γ(q) is a triple
(G, G1 , G2 ) such that one of the following occurs:
(Γ = A1 × A1 ). Now G = G1 × G2 and G1 ∼
= SU2 (q) = SU(V ) for
= G2 ∼
some Fq2 -vector space V with basis {e1 , e2 } equipped with a non-degenerate
hermitian form h for which this basis is orthonormal.
(Γ = A2 ). Now G = SU3 (q) = SU(V ) for some Fq2 -vector space V with
basis {e1 , e2 , e3 } equipped with a non-degenerate hermitian form h for which
this basis is orthonormal. As in the Curtis-Tits case, G1 (resp. G2 ) is the
stabilizer of the subspace he1 , e2 i (resp. he2 , e3 i) and the vector e3 (resp. e1 ).
We have G1 ∼
= SU2 (q).
= G2 ∼
Explicitly we have



b
 a

σ
σ
σ
σ


−b a
G1 =
: a, b ∈ Fq2 with aa + bb = 1 ,


1




 1
σ
σ


a
b : a, b ∈ Fq2 with aa + bb = 1 .
G2 =


−bσ aσ
(Γ = C2 ). Let V be an Fq2 -vector space with
be the symplectic form with Gram matrix

1

0
M =
−1 0
0 −1

basis {e1 , e2 , e3 , e4 } and let β

0
1
.


Moreover, let h be the (non-degenerate) hermitian form h for which this basis
is orthonormal.
Now G = Sp(V, β) ∩ SU(V, h) ∼
= SU2 (q) is the derived sub= Sp4 (q), G1 ∼
group of StabG (he1 , e2 i) ∩ StabG (he3 , e4 i) and G2 = StabG (e1 ) ∩ StabG (e3 ) ∼
=
Sp2 (q) ∼
= SU2 (q), . Note that Z(G) = Z(G1 ) and Z(G) ∩ G2 = {1}.

CURTIS-TITS AND PHAN AMALGAMS

9

Explicitly we have



a
b





 σ σ

−b
a
σ
σ


: a, b ∈ Fq2 with aa + bb = 1 ,
G1 = 
aσ bσ 






−b a



1







a
b
σ
σ
 : a, b ∈ Fq2 with aa + bb = 1 .
G2 = 


1






σ
σ
−b
a

Definition 2.6. For Phan amalgams, the standard identification map will be
the isomorphism g : SU2 (q) → Gi sending


a
b
−bσ aσ
to the corresponding matrix of Gi as described above.
2.4. Amalgams of Curtis-Tits and Phan type
Definition 2.7. An amalgam over a poset (P, ≺) is a collection A = {Ax |
x ∈ P} of groups, together with a collection a• = {ayx | x ≺ y, x, y ∈ P}
of monomorphisms ayx : Ax ֒→ Ay , called inclusion maps such that whenever
x ≺ y ≺ z, we have azx = azy ◦ ayx ; we shall write Ax = ayx (Ax ) ≤ Ay . A
completion of A is a group A together with a collection α• = {αx | x ∈ P}
of homomorphisms αx : Ax → A, whose images - often denoted Ax = αx (Ax )
- generate A, such that for any x, y ∈ P with x ≺ y we have αy ◦ αxy = αx .
The amalgam A is non-collapsing if it has a non-trivial completion. As a
convention, for any subgroup H ≤ AJ , let H = α(H) ≤ A.
A completion (Ã, α̃• ) is called universal if for any completion (A, α• ) there
is a unique surjective group homomorphism π : Ã → A such that α• = π ◦ α̃• .
A universal completion always exists.
Definition 2.8. Let Γ = (I, E) be a Lie diagram. A Curtis-Tits (resp. Phan)
amalgam with Lie diagram Γ over Fq is an amalgam G = {Gi , Gi,j , gi,j | i, j ∈
I} over P = {J | ∅ =
6 J ⊆ I with |J| ≤ 2} ordered by inclusion such that
for every i, j ∈ I, (Gi,j , Gi , Gj ) is a Curtis-Tits / Phan standard pair of type
Γi,j (q e ), for some e ≥ 1 as deﬁned in Subsection 2.2 and 2.3. Moreover e = 1
is realized for some i, j ∈ I. Note that in fact e is always a power of 2. This
follows immediately from connectedness of the diagram and the deﬁnition of
the standard pairs of type A2 , C2 , and 2A3 . For any subset K ⊆ I , we let
GK = {Gi , Gi,j , gi,j | i, j ∈ K}.

10

BLOK, HOFFMAN, AND SHPECTOROV

Remark 2.9. Suppose that one considers an amalgam
H = {Hi , Hi,j , hi,j | i, j ∈ I}
over Fq with diagram Γ, such that for any i, j ∈ I, the triple (Hi,j , Hi , Hj )
is not a standard pair, but there is a standard pair (Gi,j , Gi , Gj ) such that
the respective H’s are central quotients of the corresponding G’s. Then, H is
the quotient of a unique Curtis-Tits or Phan amalgam over Fq with diagram
Γ. Hence for classiﬁcation purposes it suﬃces to consider Curtis-Tits or Phan
amalgams. In particular, in view of Remark 2.4, we can restrict ourselves
to Curtis-Tits amalgams in which the only rank-2 subdiagrams are of type
A1 × A1 , A2 , C2 , and 2A3 .
+
+
Definition 2.10. Suppose G = {Gi , Gi,j , gi,j | i, j ∈ I}, G + = {G+
i , Gi,j , gi,j |
i, j ∈ I} are two Curtis-Tits (or Phan) amalgams over Fq with the same diagram Γ. Then a type preserving isomorphism φ : G → G + is a collection
φ = {φi , φi,j : i, j ∈ I} of group isomorphisms such that, for all i, j ∈ I, we
have
+
φi,j ◦ gi,j = gi,j
◦ φi
+
φi,j ◦ gj,i = gj,i
◦ φj .

Unless indicated otherwise, this is the kind of isomorphism we shall consider,
omitting the term ”type preserving”. It is also possible to consider type permuting isomorphisms, deﬁned in the obvious way.

3. Background on groups of Lie type
3.1. Automorphisms of groups of Lie type of small rank
Automorphisms of groups of Lie type are all known. In this subsection we collect some facts that we will need later on. We shall use the notation from [30].
Automorphisms of SLn (q). Deﬁne automorphisms of SLn (q) as follows (where
x = (xij )ni,j=1 ∈ SLn (q)):
cg : x 7→ xg = g −1xg
α

α : x 7→ x =
τ

(xαij )ni,j=1
t −1

τ : x 7→ x = x

(g ∈ PGLn (q)),
(α ∈ Aut(Fq )),
(transpose-inverse).



0 −1
We note that for n = 2, τ coincides with the map x 7→ x , where µ =
.
1 0
We let PΓLn (q) = PGLn (q) ⋊ Aut(Fq ).
Automorphisms of Sp2n (q). Outer automorphisms of Sp2n (q) are of the form
Aut(Fq ) as for SL2n (q), deﬁned with respect to a symplectic basis, or come
from the group GSp2n (q) ∼
= Sp2n (q).(F∗q /(F∗q )2 ) of linear similarities of the
µ

CURTIS-TITS AND PHAN AMALGAMS

symplectic form, where F∗q acts as conjugation by


λIn 0n
δ(λ) =
0 n In

11

(λ ∈ F∗q )

This only provides a true outer automorphism if λ is not a square and we ﬁnd
that PGSp2n (q) ∼
= PSp2n (q).2 if q is odd and PGSp2n (q) = PSp2n (q) if q is
even. We deﬁne
ΓSp2n (q) = GSp2n (q) ⋊ Aut(Fq )
PΓSp2n (q) = PGSp2n (q) ⋊ Aut(Fq ).
Note that, as in SL2 (q), the map τ : A → t A−1 is the inner automorphism
given by


0 n In
M=
.
−In 0n

Automorphisms of SUn (q). All linear outer automorphisms of SUn (q) are induced by GUn (q) the group of linear isometries of the hermitian form, or
are induced by Aut(Fq2 ) as for SLn (q 2 ) with respect to an orthonormal basis. The group Aut(Fq2 ) has order 2e, where q = pe , p prime. We let
ΓUn (q) = GUn ⋊ Aut(Fq2 ) and let PΓUn (q) denote its quotient over the center
(consisting of the scalar matrices). In this case, the transpose-inverse map τ
with respect to a hyperbolic basis is the composition of the inner automorphism given by


0 n In
M=
.
In 0 n

and the ﬁeld automorphism x 7→ x = xq (with respect to the hyperbolic basis).
d q2 ) of ﬁeld automorphisms of SUn (q) on a hyperbolic basis.
The group Aut(F
For ΓU2n (q) note that Aut(Fq2 ) = hαi acts with respect to an orthonormal
basis U = {u1 , . . . , u2n } for the Fq2 -vector space V with σ-hermitian form h
d q2 ) of
preserved by the group (see [30]). . We now identify a complement Aut(F
semilinear automorphisms of GU2n (q) in ΓU2n (q) with respect to a hyperbolic
basis. Fix the standard hyperbolic basis H = {ei , fi : i = 1, 2, . . . , n} so that
the elements in GU(V, h) are represented by a matrix in GU2n (q) with respect
to H. Let α ∈ Aut(Fq2 ) act on V via U. Then, Hα = {eαi , fiα : i = 1, 2 . . . , n}
is also a hyperbolic basis for V , so for some A ∈ GU2n (q), we have AH = Hα .
Now the composition α
b = A−1 ◦ α is an α-semilinear map that ﬁxes H. The
corresponding automorphism of GU2n (q) acts by applying α to the matrix
entries.
Remark 3.1. The following special case will be of particular interest when
considering a Curtis-Tits standard pairs of type 2A3 (q). In this case the action of α
b as above on SU4 (q) translates via the standard identiﬁcation maps

12

BLOK, HOFFMAN, AND SHPECTOROV

(see Deﬁnition 2.5) to actions on SL2 (q) and SL2 (q 2 ) as follows. The action
on SL2 (q 2 ) is the natural entry-wise ﬁeld automorphism action. The action on
SL2 (q) will be a product of the natural entrywise action of α
b and a diagonal
automorphism diag(f, 1), where f ∈ Fq is such that α
b(η) = f η. Note that
be translates to (left) conjugation by
NFq /Fp (f ) = −1, so in particular σ = α
diag(−1, 1) only.

Definition 3.2. Since the norm is surjective, there exists ζ ∈ Fq2 such that
NFq2 /Fq (ζ) = f −1 . We then have that diag(ζ, ζ, ζ −q, ζ −q ) ∈ GU4 (q) acts trivially on SL2 (q 2 ) and acts as left conjugation by diag(f −1 , 1) on SL2 (q). It
follows that the composition α
e of α
b and this diagonal automorphism acts entrywise as α on both SL2 (q) and SL2 (q 2 ). We now deﬁne
g q2 ) = he
αi ≤ Aut(SU4 (q)).
Aut(F

Lemma 3.3. (See [28, 30].)
(1) As Sp2 (q) = SL2 (q) ∼
= SU2 (q), we have

Aut(Sp2 (q)) = Aut(SL2 (q)) = PΓL2 (q) ∼
= PΓU2 (q) = Aut(SU2 (q)).
(2) In higher rank we have
Aut(SLn (q)) = PΓLn (q) ⋊ hτ i
Aut(Sp2n (q)) = PΓSp2n (q)
Aut(SUn (q)) = PΓUn (q)
3.1.1. Some normalizers and centralizers.



1 0
Corollary 3.4. Let G = SL3 (q). Let ϕ : SL2 (q) → G given by A 7→
0 A
and let L = im ϕ. Then,
CAut(G) (L) = hdiag(a, b, b) : a, b ∈ F∗q i ⋊ hθi.


1
where θ = τ ◦ cν : X θ 7→ t (ν −1 Xν)−1 and ν =  0 −1.
1 0

Proof This follows easily from the fact that Aut(G) ∼
= PΓL3 (q). Let τ i ◦
α ◦ cg , where cg denotes conjugation by g ∈ GL3 (q) and α ∈ Aut(Fq ). Using
transvection matrices from L over the ﬁxed ﬁeld Fαq one sees that if i = 0, then
g must be of the form diag(a, b, b), and if i = 1, then it must be of the form
diag(a, b, b)ν, for some a, b ∈ F∗q . Then, if α 6= id, picking transvections from
L with a few entries in Fq − Fαq one veriﬁes that α must be the identity.


CURTIS-TITS AND PHAN AMALGAMS

13

4. Classification of Curtis-Tits amalgams
4.1. Fundamental root groups in Curtis-Tits standard pairs
Lemma 4.1. Let q be a power of the prime p. Suppose that (G, G1 , G2 ) is a
Curtis-Tits standard pair of type Γ(q) as in Subsection 2.2. For {i, j} = {1, 2},
let Sj = Sylp (Gj ).
(1) There exist two groups Xi,ε
j ∈ Sj (ε = +, −) such that for any X ∈ Sj
we have
hGi , Xi ≤ Pεi if and only if X = Xi,ε
j
−
where P+
i and Pi are the two parabolic subgroups of G containing Gi .
If X 6= Xi,ε
j , then
(
(Gi × Gxi ) ⋊ hxi if Γ(q) = C2 (2),
hGi , Xi =
G
else.

where in the C2 (2) case X = hxi.
(2) We can select the signs ε so that Xj,ε
commutes with Xji,−ε , but not
i
i,ε
j,ε
with Xi,ε
j and, in fact hXi , Xj i is contained in the unipotent radical
Uεi,j of a unique Borel subgroup of Gi,j , namely Bεi,j = Pεi ∩ Pεj .
Proof We ﬁrst prove part 1. by considering all cases.
A2 (q), q ≥ 3. View G = SL3 (q) = SL(V ) for some Fq -vector space with basis
{e1 , e2 , e3 }. By symmetry we may assume that i = 1 and j = 2. Let G1
(resp. G2 ) stabilize he1 , e2 i and ﬁx e3 (resp. stabilize he2 , e3 i) and ﬁx e1 ). A
root group in G2 is of the form Xv = StabG2 (v) for some v ∈ he2 , e3 i. We let
−
ε
ε
X+
2 = Xe2 and X2 = Xe3 . It clear that for ε = + (resp. ε = −) hG1 , X2 i = P
is contained in (but not equal to) the parabolic subgroup stabilizing he1 , e2 i
(resp. he3 i). Now suppose that X ∈ S2 is diﬀerent from Xε2 (ε = +, −)
and X = Xλe2 +e3 for some λ ∈ F∗q . Consider the action of a torus element
d = diag(µ, µ−1, 1) ∈ G1 by conjugation on G2 . Then Xd = Xµλe2 +e3 . Since
|Fq | ≥ 3, Xd 6= X for some d and so we have
(4.1)

hGi , Xi ≥ hGi , X, Xdi = hGi , Gj i = G.

−
A2 (2). In this case S2 = {X+
2 , X = hri, X2 }, where r is the Coxeter element
ﬁxing e1 and interchanging e2 and e3 . It follows that Gr1 is the stabilizer of
the subspace decomposition he2 i ⊕ he1 , e3 i and hence hG1 , Xi = G.
C2 (q), q ≥ 3, X short root. We use the notation of Subsection 2.2. First,
let i = 2, j = 1, let G2 ∼
= SL2 (q) be the stabilizer of e1 and e3
= Sp2 (q) ∼
∼
and let G1 = SL2 (q) be the derived subgroup of the stabilizer of the isotropic
2-spaces he1 , e2 i and he3 , e4 i. Root groups in G1 are of the form Xu,v =
StabG1 (u) ∩ StabG1 (v), where u ∈ he1 , e2 i and v ∈ he3 , e4 i are orthogonal. Let

14

BLOK, HOFFMAN, AND SHPECTOROV

−
X+
1 = Xe1 ,e4 and X1 = Xe2 ,e3 . It is easy to verify that for ε = + (resp. ε = −)
ε
ε
hG2 , X1 i = P is contained in the parabolic subgroup stabilizing he1 i (resp.
he3 i). Now let X = Xe1 +λe2 ,e3 −λ−1 e4 for some λ ∈ F∗q . Consider the action
of a torus element d = diag(1, µ−1 , 1, µ) ∈ G2 by conjugation on G1 . Then
Xd = Xhe1 +λµe2 i,e3 −λ−1 µ−1 e4 . Since q ≥ 3, Xd 6= X for some d and so, for i = 1,
and these G2 , X and d, we have (4.1) again.
C2 (q), q ≥ 4, X long root. Now, we let i = 1 and j = 2. Root groups in
G2 are of the form Xu = StabG2 (u) where u ∈ he2 , e4 i. Let X+
2 = Xe2 and
ε
ε
X−
=
X
.
It
is
easy
to
verify
that
for
ε
=
+
(resp.
ε
=
−)
hG
e4
1 , X2 i = P is
2
contained in the parabolic subgroup stabilizing he1 , e2 i (resp. he1 , e4 i). Now
let X = Xe2 +λe4 for some λ ∈ F∗q .
Consider the action of a torus element d = diag(µ, µ−1, µ−1 , µ) ∈ G1 by
conjugation on G2 . Then Xd = Xµe2 +µ−1 λe4 . Now if q ≥ 4, then Xd 6= X for
some d and so so, for these G1 , X and d, we have (4.1) again.
C2 (q), q = 3, X long root. The proof for the case q ≥ 4 does not yield the
result since, for q = 3, the element d centralizes G2 . A direct computation
in GAP shows that the conclusion still holds, though. Let x ∈ X = Xe2 +e4
send e2 to e4 . Then G1 and Gx1 contains two short root groups ﬁxing e1 and
e3 . Their commutators generate a long root group ﬁxing e1 , e2 , and e4 , while
being transitive on the points he3 + λe1 i. Further conjugation with an element
in G1 interchanging the points he1 i and he2 i yields a long root group in G2
diﬀerent from X and we obtain an equation like (4.1) again.
C2 (2). First note that G ∼
= Sp4 (2) ∼
= O5 (2) is self point-line dual, so we
only need to consider the case where G2 = StabG (e1 ) ∩ StabG (e3 ) and G1 =
−
StabG (he1 , e2 i) ∩ StabG (he3 , e4 i). Now S1 = {X+
1 , X1 , hxi}, where x is the
permutation matrix of (1, 2)(3, 4). The conclusion follows easily.
2
A3 (q). We use the notation of Subsection 2.2. First, let i = 2, j = 1, let G2 ∼
=
2
∼
∼
SU2 (q) = SL2 (q) be the stabilizer of e1 and e3 and let G1 = SL2 (q ) be the
derived subgroup of the simultaneous stabilizer in G = SU4 (q) of the isotropic
2-spaces he1 , e2 i and he3 , e4 i. Root groups in G1 are of the form Xu,v =
StabG1 (u) ∩ StabG1 (v), where u ∈ he1 , e2 i and v ∈ he3 , e4 i are orthogonal. Let
−
X+
1 = Xe1 ,e4 and X1 = Xe2 ,e3 . It is easy to verify that for ε = + (resp. ε = −)
hG2 , Xε1 i = Pε is contained in the parabolic subgroup stabilizing he1 i (resp.
he3 i). Now let X = Xe1 +λe2 ,e3 −λ−σ e4 for some λ ∈ F∗q2 . Consider the action
of a torus element d = diag(1, µ−1, 1, µ) ∈ G2 (with µ ∈ F∗q ) by conjugation
on G1 . Then Xd = Xe1+λµe2 ,e3 −λ−σ µ−1 e4 . There are q − 1 choices for µ, so if
q ≥ 3, then Xd 6= X for some d. Hence, for i = 1, and these G2 , X and d, we
have (4.1) again. The case q = 2 is a quick GAP calculation.
Now, we let i = 1 and j = 2. Root groups in G2 are of the form Xu =
−
StabG2 (u) where u ∈ he2 , e4 i is isotropic. Let X+
2 = Xe2 and X2 = Xe4 . It is
easy to verify that for ε = + (resp. ε = −) hG1 , Xε2 i = Pε is contained in the

CURTIS-TITS AND PHAN AMALGAMS

15

parabolic subgroup stabilizing he1 , e2 i (resp. he1 , e4 i). Now let X = Xe2 +λe4
for some λ ∈ F∗q2 where Tr(λ) = λ + λσ = 0. Consider the action of a torus
element d = diag(µ, µ−1, µ−σ , µσ ) ∈ G1 (for some µ ∈ F∗q2 ) by conjugation
on G2 . Then Xd = Xµe2 +µ−σ λe4 . The q 2 − 1 choices for µ result in q − 1
diﬀerent conjugates. Thus, if q − 1 ≥ 2, then Xd 6= X for some d and so so,
for these G1 , X and d, we have (4.1) again. The case q = 2 is a quick GAP
calculation. Namely, in this case, X = hxi, where x is the only element of
−
order 2 in G2 ∼
= S3 that does not belong to X+
1 ∪ X1 ; it is the Coxeter element
that ﬁxes e1 and e3 and interchanges e2 and e4 . Now hG1 , Gr1 i contains the
long root group generated by the commutators of the short root group ﬁxing
e1 in G1 and Gr1 , and likewise for e2 , e3 , and e4 . In particular, we have
(4.2)

hG1 , Xi ≥ hG1 , Gr1 i ≥ hG1 , G2 i = G

We now address part 2. Note that the positive and negative fundamental
j,ε
−
root groups with respect to the torus B+
i,j ∩ Bi,j satisfy the properties of Xi
and Xi,ε
j so by the uniqueness statement in 1. they must be equal. Now the
claims in part 2. are the consequences of the Chevalley commutator relations.

Remark 4.2.
−
Explicitly, the groups {X+
i , Xi } (i = 1, 2), possibly
for the Curtis-Tits standard pairs are as follows.
For Γ = A2 , we have




 1 b

 1
+
−


0 1
X1 =
: b ∈ Fq , and X1 =  c



1




 1

 1
+
−


1 b : b ∈ Fq , and X2 = 
X2 =



0 1

up to a switch of signs,

0
1
1



 : c ∈ Fq










,

1 0  : c ∈ Fq .

c 1

For Γ = C2 , we have






1
0
1
b














c
1
0
1
−
+




,
:
c
∈
F
,
and
X
=
:
b
∈
F
X1 = 
q
q
1

1 −c
1 0












0 1
−b 1






1
1














1
0
1
b
−
+




: c ∈ Fq .
: b ∈ Fq , and X2 = 
X2 = 
1 
1 












c
1
0
1

16

BLOK, HOFFMAN, AND SHPECTOROV

For Γ = 2A3 , we have (with η ∈ Fq2 of trace 0),






1
b
1
0














0
1
c
1
+
−
 : b ∈ Fq2 , and X1 = 
 : c ∈ Fq 2 ,
X1 = 


1 0
1 −c












−b 1
0 1






1
1














1
0
1
bη
−
+




: c ∈ Fq .
X2 = 
 : b ∈ Fq  , and X2 = 
1 
1










cη −1
1
0
1
4.2. Weak systems of fundamental groups

In this subsection we show that a Curtis-Tits amalgam with 3-spherical diagram determines a collection of subgroups of the vertex groups, called a weak
system of fundamental root groups. We then use this to determine the coefﬁcient system of the amalgam in the sense of [2], which, in turn is applied to
classify these amalgams up to isomorphism.
Definition 4.3. Suppose that G = {Gi , Gi,j , gi,j | i, j ∈ I} is a CT amalgam.
−
For each i ∈ I let X+
i , Xi ≤ Gi be a pair of opposite root groups. We say
−
that {X+
i , Xi | i ∈ I} is a weak system of fundamental root groups if, for any
−
edge {i, j} ∈ E there are opposite Borel groups B+
i,j and Bij in Gi,j , each of
+
−
which contains exactly one of {Xi , Xi }.
We call G orientable if we can select Xεi , Bεij (ε = +, −) for all i, j ∈ V such
ε
ε
that Xi , Xj ≤ Bεij . If this is not possible, we call G non-orientable.
The relation between root groups and Borel groups is given by the following
well-known fact.
Lemma 4.4. Let q be a power of the prime p. Let G be a universal group of
Lie type Γ(q) and let X be a Sylow p-subgroup. Then, NG (X) is the unique
Borel group B of G containing X.
Proposition 4.5. Suppose that G = {Gi , Gi,j , gi,j | i, j ∈ I} is a CT amalgam
with connected 3-spherical diagram Γ. If G has a non-trivial completion (G, γ),
then it has a unique weak system of fundamental root groups.
Proof We ﬁrst show that there is some weak system of fundamental root
groups. For every edge {i, j}, let Xi,ε
j be the groups of Lemma 4.1. Suppose
that there is some subdiagram ΓJ with J = {i, j, k} in which j is connected
i,−
k,−
to both i and k, such that {Xi,+
6 {Xk,+
j , Xj } =
j , Xj } as sets. Without
loss of generality assume that Γi,j = A2 (by 3-sphericity) and moreover, that
i,−
Xk,+
6∈ {Xi,+
j
j , Xj }. For any subgroup H of a group in G , write H = γ(H).
Now note that Xkj,− commutes with Xjk,+ and since Γ contains no triangles it

CURTIS-TITS AND PHAN AMALGAMS

17

also commutes with Gi . But then Xkj,− commutes with hXjk,+ , Gi i which, by
Lemma 4.1, equals Gi,j (this is where we use that Γi,j = A2 ), contradicting that
Xkj,− does not commute with Xjk,− ≤ Gi,j . Thus, if there is a completion, then
by connectedness of Γ, for each i ∈ I we can pick a j ∈ I so that {i, j} ∈ E
j,±
and set X±
and drop the superscript. We claim that {X±
i = Xi
i | i ∈ I} is
a weak system of fundamental root groups. But this follows from part 2. of
Lemma 4.1.
The uniqueness derives immediately from the fact that by Lemma 4.1,
i,−
gi,j (Xi,+
j ) and gj,i (Xj ) are the only two Sylow p-subgroups in gj,i (Gj ) which
do not generate Gi,j with gi,j (Gi ).

An immediate consequence of the results above is the following observation.
Corollary 4.6. Suppose that G = {Gi , Gi,j , gi,j | i, j ∈ I} is a CT amalgam
with connected 3-spherical diagram Γ. Then, an element of NAut(Gi,j ) (Gi , Gj )
+
−
+
−
−
either fixes each of the pairs (Xi , Xi ), (Xj , Xj ), and (B+
i,j , Bi,j ) or it reverses
each of them. In particular,
+

−

+

−

NAut(Gi,j ) (Gi , Gj ) = NAut(Gi,j ) ({Xi , Xi }) ∩ NAut(Gi,j ) ({Xj , Xj }).
4.3. The coefficient system of a Curtis-Tits amalgam
The automorphisms of a Curtis-Tits standard pair will be crucial in the classiﬁcation of Curtis-Tits amalgams and we will need some detailed description
of them.
We now ﬁx a Curtis-Tits amalgam G = {Gi , Gi,j , gi,j | i, j ∈ I} of type Γ(q),
where for every i, j ∈ I, gi,j is the standard identiﬁcation map of Deﬁnition 2.5.
−
Then, G has a weak system of fundamental root groups X = {{X+
i , Xi } : i ∈
I} as in Subsection 4.1.
Remark 4.7. Let G = {Gi , Gi,j , gi,j | i, j ∈ I} be a Curtis-Tits amalgam
over Fq with given diagram Γ. Next suppose that Γ is connected 3-spherical,
and that G and G are non-collapsing. Then, by Proposition 4.5, G and G each
have a weak system of fundamental root groups. Now note that for each i ∈ I,
Aut(Gi ) is 2-transitive on the set of Sylow p-subgroups. Thus, for each i ∈ I
and all j ∈ I − {i}, we can replace gi,j by gi,j ◦ αi , to form a new amalgam
isomorphic to G , whose weak system of fundamental root groups is exactly X .
Thus, in order to classify non-collapsing Curtis-Tits amalgams over Fq with
diagram Γ up to isomorphism, it suﬃces to classify those whose weak system
of fundamental root groups is exactly X .
Definition 4.8. Suppose that G = {Gi , Gi,j , gi,j | i, j ∈ I} is a Curtis-Tits
amalgam over Fq with connected 3-spherical diagram Γ. Denote the associated
−
weak system of fundamental root groups as X = {{X+
i , Xi } : i ∈ I}. The

18

BLOK, HOFFMAN, AND SHPECTOROV

coefficient system associated to G is the collection A = {Ai , Ai,j , ai,j | i, j ∈ I}
where, for any i, j ∈ I we set
−
Ai = NAut(Gi ) ({X+
i , Xi }),
ε

ε

Ai,j = NAut(Gi,j ) ({Xi : ε = +, −}) ∩ NAut(Gi,j ) ({Xj : ε = +, −}),
−1
ai,j : Ai,j → Aj is given by restriction: ϕ 7→ gj,i
◦ ρi,j (ϕ) ◦ gj,i .

where ρi,j (ϕ) is the restriction of ϕ to Gj ≤ Gi,j .
From now on we let A be the coeﬃcient system associated to G . The
signiﬁcance for the classiﬁcation of Curtis-Tits amalgams with weak system of
fundamental root groups is as follows:
Proposition 4.9. Suppose that G and G + are Curtis-Tits amalgams with
diagram Γ over Fq with weak system of fundamental root groups X .
+
+
(1) For all i, j ∈ I, we have gi,j = gi,j ◦ δi,j and gi,j
= gi,j ◦ δi,j
for some
+
δi,j , δi,j ∈ Ai ,
(2) For any isomorphism φ : G → G + and i, j ∈ I, we have φi ∈ Ai ,
+
−1
φ{i,j} ∈ Ai,j , and ai,j (φ{i,j}) = δi,j
◦ φi ◦ δi,j
.
−1
Proof Part 1. follows since, for any i, j ∈ I we have gi,j
◦ gi,j ∈ Aut(G1 ) and
+
−
−
{gi,j (X+
i ), gi,j (Xi )} = {g i,j (Xi ), g i,j (Xi )}.

Part 2. follows from Corollary 4.6 since, for any i, j ∈ I,
+
+
(Gi,j , gi,j (Gi ), gj,i (Gj )) = (Gi,j , gi,j (Gi ), gj,i (Gj )) = (Gi,j , gi,j
(Gi ), gj,i
(Gj )).


We now determine the groups appearing in the coeﬃcient system A associated to G .
Lemma 4.10. Fix i ∈ I and let q be such that Gi ∼
= SL2 (q). Then,
A i = Ti ⋊ Ci ,
where Ti is the subgroup of diagonal automorphisms in PGL2 (q) and Ci =
hτ, Aut(Fq )i.
Proof This follows from the fact that via the standard embedding map gi,j
−
the groups X+
i and Xi of the weak system of fundamental root groups are the
subgroups of unipotent upper and lower triangular matrices in SL2 (q).

Lemma 4.11. Let A be the coefficient system associated to the standard
Curtis-Tits amalgam G of type Γ(q) and the weak system of fundamental root
groups X .

CURTIS-TITS AND PHAN AMALGAMS

19

If Γ = A1 × A1 , we have Gi,j = Gi × Gj , gi,j and gj,i are identity maps,
and
(4.3)
Ai,j = Ai × Aj ∼
= Ti,j ⋊ Ci,j .
where Ti,j = Ti × Tj and Ci,j = Ci × Cj . Otherwise,
Ai,j = Ti,j ⋊ Ci,j ,
where
(4.4)

Ci,j

(
Aut(Fq ) × hτ i
= g
Aut(Fq2 ) × hτ i

for Γ = A2 , C2
for Γ = 2A3

and Ti,j denotes the image of the standard torus T in Aut(Gi,j ). Note that

∗

if Γ = A2
hdiag(a, b, c) : a, b, c ∈ Fq i ≤ GL3 (q)
−1
−1
∗
T = hdiag(ab, a b, a , a) : a, b ∈ Fq i ≤ GSp4 (q) if Γ = C2

hdiag(a, b, a−q , b−q ) : a, b ∈ F∗ i ≤ GU (q)
if Γ = 2A3 .
4
q2

Remark 4.12. Remarks on Lemma 4.11
(1) We view Sp2n (q) and SU2n (q) as a matrix group with respect to a
symplectic (resp. hyperbolic) basis for the 2n-dimensional vector space
d q2 )) acts entrywise on the matrices.
V and Aut(Fq ) (resp. Aut(F
(2) The map τ is the transpose-inverse map of Subsection 3.1.
(3) Recall that in the 2A3 case, Remark 3.1 and Deﬁnition 3.2 describe the
g q2 ) ≤ Ci,j on Gi and Gj via the standard identiﬁcation
actions of Aut(F
maps.

Proof We ﬁrst consider the A1 × A1 case of (4.3). When Γ = A1 × A1 , then
±
Gi,j = Gi × Gj and since the standard root groups Xi generate Gi (i = 1, 2),
their simultaneous normalizer must also normalize Gi and Gj . Thus the claim
follows from Lemma 4.10.
We now deal with all remaining cases simultaneously. In the 2A3 case we note
d q2 )
g q2 ) ≤ Ti,j ⋊Aut(F
that from Remark 3.1 and Deﬁnition 3.2 we see that Aut(F
is simply a diﬀerent complement to Ti,j , so it suﬃces to prove the claim with
d q2 ).
g q2 ) replaced by Aut(F
Aut(F
−
Consider the descriptions of the set {X+
i , Xi } in all cases from Subsection 4.1. We see that since τ acts by transpose-inverse, it interchanges X+
i
and X−
i for i = 1, 2 in all cases, hence it also interchanges positive and negative Borel groups (see Corollary 4.6). Thus it suﬃces to consider those automorphisms that normalize the positive and negative fundamental root groups.
d q2 )) act entrywise, they
Since all ﬁeld automorphisms (of Aut(Fq ) and Aut(F
do so. Clearly so does T. Thus we have established ⊇.

20

BLOK, HOFFMAN, AND SHPECTOROV

We now turn to the reverse inclusion. By Lemma 3.3 and the description
of the automorphism groups in Subsection 3.1 any automorphism of Gi,j is a
product of the form gατ i where g is linear, α is a ﬁeld automorphism (from
d q2 ) in the 2A3 case) and i = 0, 1. As we saw above τ and α preserve the
Aut(F
root groups, so it suﬃces to describe g in case it preserves the sets of opposite
root groups. A direct computation shows that g must be in T.

Next we describe the connecting maps ai,j of A .
Lemma 4.13. Let A be the coefficient system of the standard Curtis-Tits
amalgam G over Fq with diagram Γ and weak system of fundamental root
groups X . Fix i, j ∈ I and let (Gi,j , Gi , Gj ) be a Curtis-Tits standard pair in
G with diagram Γi,j . Denote a = (aj,i , ai,j ) : Ai,j → Ai × Aj . Then, we have
the following:
(1) If Γi,j = A1 × A1 , then a is an isomorphism inducing Ti,j ∼
= Ti × Tj
and Ci,j ∼
C
×
C
.
= i
j
(2) If Γi,j = A2 , or 2A3 , then a : Ti,j → Ti × Tj is bijective.
∼
=
(3) If Γi,j = C2 , then a : Ti,j → T2i × Tj has index 1 or 2 in Ti × Tj
depending on whether q is even or odd.
(4) If Γi,j = A2 or C2 , then a : Ci,j → Ci ×Cj is given by τ s α 7→ (τ s α, τ s α)
(for s ∈ {0, 1} and α ∈ Aut(Fq )) which is a diagonal embedding.
(5) If Γ = 2A3 , then a : Ci,j → Ci × Cj , is given by τ s α
er 7→ (τ s αr , τ s αr )
e 7→ (σ, id).
(for s ∈ {0, 1}, r ∈ N, and α : x 7→ xp for x ∈ Fq2 . Here σ
Remark 4.14.
(1) In 4. τ acts as transpose-inverse and α acts entry-wise
on Gi,j , Gi and Gj .

Proof 1. This is immediate from Lemma 4.11.
For the remaining cases, recall that for any ϕ ∈ Ai,j , we have ai,j : ϕ 7→ g−1
◦
j,i
ρi,j (ϕ) ◦ gj,i , where ρi,j (ϕ) is the restriction of ϕ to Gj ≤ Gi,j (Deﬁnition 4.8)
and gi,j is the standard identiﬁcation map of Deﬁnition 2.5. Note that for
Γi,j = A2 , C2 the standard identiﬁcation map transforms the automorphism
ρj,i (ϕ) of Gi essentially to the “same” automorphism ϕ of Gi , whereas for
Γi,j = 2A3 , we must take Remark 3.1 into account.
2. Let Γi,j = A2 . Every element of Ti,j (Ti , and Tj respectively) is given
by a unique matrix of the form diag(a, 1, c) (diag(a, 1), and diag(1, c)), and we
have
(aj,i , ai,j ) : diag(a, 1, c) 7→ (diag(a, 1), diag(1, c))

(a, c ∈ F∗q ),

which is clearly bijective. In the 2A3 case, every element of Ti,j (Ti , and Tj respectively) is given by a unique matrix of the form diag(ab−1 , 1, a−q b−1 , b−(q+1) )

CURTIS-TITS AND PHAN AMALGAMS

21

(diag(1, b−(q+1) ), and diag(ab−1 , 1)), and we have
a : diag(ab−1 , 1, a−q b−1 , b−(q+1) )7→(diag(1, b−(q+1) ), diag(ab−1 , 1)).
This map is onto since NFq2 /Fq : b 7→ bq+1 is onto. Its kernel is trivial, as it is
given by pairs (a, b) ∈ Fq2 , with a = b and bq+1 = 1 so that also a−q b−1 = 1.
3. In the C2 -case, every element of Ti,j is given by a unique diagonal matrix
diag(a2 b, b, 1, a2 ) (a, b ∈ Fq ). Every element of Ti (resp. Tj ) is given by a
unique diag(c, 1) (resp. diag(d, 1)). Now we have
a : diag(a2 b, b, 1, a2 ) 7→ (diag(a2 , 1), diag(ba−2 , 1)).
It follows that a is injective and has image T2i × Tj . The rest of the claim
follows.
4. The ﬁeld automorphism α ∈ Aut(Fq ) acts entrywise on the matrices in
Gi,j = SL3 (q), or Sp4 (q), and Gi = Gj = SL2 (q). In the case G = Sp4 (q),
we saw in Subsection 3.1 that τ is inner and coincides with conjugation by
M. This clearly restricts to conjugation by µ on both G2 = Sp2 (q) and
G1 = SL2 (q), which is again τ . Clearly these actions correspond to each other
via the standard identiﬁcation maps gi,j and gj,i .
g q2 ) ≤ Ci,j on Gi and Gj via a was explained in
5. The action of Aut(F
Remark 3.1 and Deﬁnition 3.2. In case Gi,j = SU4 (q), τ is given by conjugation
by M composed with the ﬁeld automorphism σ
b, where σ : x 7→ xq for x ∈ Fq2 .
The same holds for Gj = SU2 (q) and τ restricts to Gi = SL2 (q 2 ) as transposeinverse. In view of Remark 3.1 we see that via the standard identiﬁcation map
each restricts to transpose inverse on Gi and Gj .

4.4. A standard form for Curtis-Tits amalgams
Suppose that G = {Gi , Gi,j , gi,j | i, j ∈ I} is a Curtis-Tits amalgam over Fq
with 3-spherical diagram Γ. Without loss of generality we will assume that all
inclusion maps gi,j are the standard identiﬁcation maps of Deﬁnition 2.5.
By Proposition 4.5 it possesses a weak system of fundamental root groups
−
X = {{X+
i , Xi } : i ∈ I},

which via the standard embeddings gi,j can be identiﬁed with those given in
Subsection 4.1 (note that orienting X may involve changing some signs). Let
A = {Ai , Ai,j , ai,j | i, j ∈ I} be the coeﬃcient system associated to G and X .
We wish to classify all Curtis-Tits amalgams G = {Gi , Gi,j , gi,j | i, j ∈ I}
over Fq with the same diagram as G with weak system of fundamental root
groups X up to isomorphism of Curtis-Tits amalgams. By Proposition 4.9
we may restrict to those amalgams whose connecting maps are of the form
gi,j = gi,j ◦ δi,j for δi,j ∈ Ai for all i ∈ I.

22

BLOK, HOFFMAN, AND SHPECTOROV

Definition 4.15. The trivial support of G (with respect to G ) is the set
{(i, j) ∈ I × I | gi,j = gi,j } (that is, δi,j = idGi in the notation of Proposition 4.9). The word “trivial” derives from the assumption that the gi,j ’s are
the standard identiﬁcation maps of Deﬁnition 2.5.
Fix some spanning tree Σ ⊆ Γ and suppose that E − E Σ = {{is , js } : s =
1, 2, . . . , r} so that H1 (Γ, Z) ∼
= Zr .
Proposition 4.16. There is a Curtis-Tits amalgam G (Σ) over Fq with the
same diagram as G and the same X , which is isomorphic to G and has the
following properties:
(1) G has trivial support S = {(i, j) ∈ I × I | {i, j} ∈ E Σ} ∪ {(is , js ) : s =
1, 2 . . . , r}.
(2) for each s = 1, 2, . . . , r, we have gjs ,is = gj ,i ◦ γjs ,is , where γjs,is ∈ Cjs .
s s

Lemma 4.17. There is a Curtis-Tits amalgam G + over Fq with the same
diagram as G and the same X , which is isomorphic to G and has the following
properties: For any u, v ∈ I, if gu,v = gu,v ◦ γu,v ◦ du,v , for some γu,v ∈ Cu and
+
du,v ∈ Tu , then gu,v
= gu,v ◦ γu,v .
Proof Note that we have |I| ≥ 2 and that Γ is connected. Fix u ∈ I. Since Γ
is 3-spherical, there is at most one w ∈ I such that (Gu,w , Gu , Gw ) is a CurtisTits standard pair of type B2 or C2 . If there is no such w, let w be an arbitrary
+
vertex such that {u, v} ∈ E Γ. We deﬁne G + by setting gu,v
= gu,v ◦ γu,v for
all v 6= u.
Next we deﬁne φ : G → G + setting φu = du,w and φv = idGv for all v 6= u.
Now note that setting φu,w = idGu,v , {φu,w , φu , φw } is an isomorphism of the
+
subamalgams of G{u,w} and G{u,w}
. As for φu,v for v 6= w, note that in order for
+
{φu,v , φu , φv } to be an isomorphism of the subamalgams of G{u,v} and G{u,v}
,
we must have
+
gu,v
φu = φu,v ◦ gu,v
+
gv,u
φv = φu,v ◦ gv,u

which translates as
gu,v ◦ γu,v ◦ du,w = φu,v ◦ gu,v ◦ γu,v ◦ du,v
gv,u ◦ δv,u = φu,v ◦ gv,u ◦ δv,u
or in other words
−1
γu,v ◦ du,w ◦ d−1
u,v ◦ γu,v = av,u (φu,v )

idGv = au,v (φu,v ).

CURTIS-TITS AND PHAN AMALGAMS

23

−1
Note that γu,v ◦ du,w ◦ ◦d−1
u,v ◦ γu,v ∈ Tu ✁ Au . Now by Lemma 4.13 as
(Gu,v , Gu , Gv ) is not of type B2 or C2 the map (aj,i , ai,j ) : Ti,j → Ti × Tj
is onto. In particular, the required φu,v ∈ Tu,v can be found. This completes
the proof.


By Lemma 4.17 in order to prove Proposition 4.16 we may now assume that
gu,v = gu,v ◦ γu,v for some γu,v ∈ Cu for all u, v ∈ I.
Let G = {Gi , Gj , Gi,j , gi,j = gi,j ◦ γi,j | i, j ∈ I} be a Curtis-Tits amalgam
over Fq with |I| = 2 and γi,j ∈ Ci and γj,i ∈ Cj . We will describe all possible
+
+
+
amalgams G + = {Gi , Gj , Gi,j , gi,j
= gi,j ◦ γi,j
| i, j ∈ I} with γi,j
∈ Ci and
+
γj,i ∈ Cj , isomorphic to G via an isomorphism φ with φi ∈ Ci , φj ∈ Cj and
φi,j ∈ Ci,j .
Gi,j

g i, j

gj

,i

Gi

Gj

φi

φi,j

φj

Gi

Gj

g+
i, j

+

Gi,j

g j,i

Figure 1. The commuting hexagon of Corollary 4.18.
+
Corollary 4.18. With the notation introduced above, fix the maps γi,j , γi,j
, φi ∈
+
Ci as well as γj,i ∈ Cj . Then for any one of γj,i , φj ∈ Cj , there exists a choice
γ ∈ Ci for the remaining map in Cj so that there exists φi,j making the diagram in Figure 1 commute. Moreover, if Γi,j is one of A2 , B2 , C2 , 2A3 , then
γ is unique, whereas if Γi,j = 2D3 , then there are exactly two choices for γ.

Proof The ﬁrst claim follows immediately from the fact that the restriction
maps aj,i : Ci,j → Ci and ai,j : Ci,j → Cj in part 4. and 5. of Lemma 4.13 are
both surjective. The second claim follows from the fact that aj,i : Ci,j → Ci is
injective except if Γi,j = 2D3 in which case it has a kernel of order 2.

Proof (of Proposition 4.16) By Lemma 4.17 we may assume that gi,j = gi,j ◦
γi,j for some γi,j ∈ Ci for all i, j ∈ I.
For any (possibly empty) subset T ⊆ V let S(T ) be the set of pairs (i, j) ∈ S
such that i ∈ T . Clearly the trivial support of G contains S(∅).

24

BLOK, HOFFMAN, AND SHPECTOROV

We now show that if T is the vertex set of a (possibly empty) proper subtree
of Σ, and u is a vertex such that T ∪ {u} is also the vertex set of a subtree of
Σ, then for any Curtis-Tits amalgam G whose trivial support contains S(T ),
there is a Curtis-Tits amalgam G + isomorphic to G , whose trivial support
contains S(T ∪ {u}).
Once this is proved, Claim 1. follows since we can start with T = ∅ and
end with a Curtis-Tits amalgam, still isomorphic to G , whose trivial support
contains S.
Now let T and u be as above. We ﬁrst deal with the case where T 6= ∅.
Let t be the unique neighbor of u in the subtree of Σ with vertex set T ∪ {u}.
+
+
We shall deﬁne an amalgam G + = {Gi , Gi,j , gi,j
= gi,j ◦ γi,j
| i, j ∈ I} and an
+
+
isomorphism φ : G → G , where γi,j , φi ∈ Ci and φ{i,j} ∈ Ci,j for all i, j ∈ I.
+
First note that it suﬃces to deﬁne gi,j
, φi and φ{i,j} for {i, j} ∈ E: given this
data, by the A1 × A1 case in Lemma 4.11 and Lemma 4.18, for any non-edge
{k, l} there is a unique φ{k,l} ∈ Ck,l such that (φk,l , φk , φl ) is an isomorphism
+
between G{k,l} and G{k,l}
.
Before deﬁning inclusion maps on edges, note that since Γ is 3-spherical,
no two neighbors of u in Γ are connected by an edge. Therefore we can
unambiguously set
+
gi,j
= gi,j for u 6∈ {i, j} ∈ E Γ.
+
+
Note that both maps gt,u
and gu,t
are forced upon us, but at this point for any
+
+
other neighbor v of u, only one of gu,v
and gv,u
is forced upon us. We set
+
gt,u
= gt,u , and
+
gv,u
= gv,u for v ∈ I with (u, v) 6∈ S and (v, u) ∈ S.

To extend the trivial support as required, we set
+
gu,v
= gu,v for v ∈ I with (u, v) ∈ S.

We can already specify part of φ: Set
φi = idGi for i ∈ I − {u},
φ{i,j} = idGi,j for u 6∈ {i, j} ∈ E Γ.
Thus, what is left to specify is the following: φu and φ{u,t} and, for all neighbors
v 6= t of u we must specify φ{u,v} as well as
+
gu,v
if (u, v) 6∈ S,
+
gv,u
if (u, v) ∈ S.

Figures 2 and 3 describe the amalgam G (top half) and G + (bottom half) at
the vertex u, where t ∈ V Σ, v ∈ V Γ, and {u, t}, {u, v} ∈ E Γ. Inclusion maps

CURTIS-TITS AND PHAN AMALGAMS

25

from G + forced upon us are indicated in bold, the dotted arrows are those we
must deﬁne so as to make the diagram commute.
Gt,u

g t,u

gu

Gu,v

g u, v

,t

Gt

gv

,u

Gu

φt := idGt

φu,t

Gv

φu

G+
t

φv := idGv

φu,v

G+
u

g

t,u

G+
t,u

G+
v

g

g u, t

u, v

+

G+
u,v

g v ,u

Figure 2. The case (u, v) ∈ S and (v, u) 6∈ S.

Gt,u

g t,u

gu

Gu,v

g u, v

,t

Gt

gv

,u

Gu

φt := idGt

φu,t

Gv

φu

G+
t

φv := idGv

φu,v

G+
u

g

t,u

G+
t,u

g u, t

G+
v

g+
u, v

G+
u,v

g v ,u

Figure 3. The case (u, v) 6∈ S and (v, u) ∈ S.
In these ﬁgures all non-dotted maps are of the form gi,j ◦ γi,j for some
γi,j ∈ Ci hence we can ﬁnd the desired maps using Corollary 4.18.
In case T = ∅, the situation is as described in Figures 2 and 3 after removing
the {u, t}-hexagon and any conditions it may impose on φu , and letting v run
over all neighbors of u. That is, we must now deﬁne φu , and for any neighbor
v of u, we must ﬁnd φu,v as well as
+
gu,v
if (u, v) 6∈ S,
+
gv,u
if (u, v) ∈ S.

26

BLOK, HOFFMAN, AND SHPECTOROV

To do so we let φu = idGu ∈ Cu . Finally, for each neighbor v of u we simply
+
+
let gu,v
= gu,v (so that φu,v = idGi,j ∈ Ci,j ) if (u, v) 6∈ S, and we obtain gv,u
and φu,v ∈ Cu,v using Corollary 4.18 if (u, v) ∈ S.

4.5. Classification of Curtis-Tits amalgams with 3-spherical diagram
In the case where G is a Curtis-Tits amalgam over Fq whose diagram is a
3-spherical tree, Proposition 4.16 says that G ∼
= G.
Theorem 4.19. Suppose that G is a Curtis-Tits amalgam with a diagram that
is a 3-spherical tree. Then, G is unique up to isomorphism. In particular any
Curtis-Tits amalgam with spherical diagram is unique.
Lemma 4.20. Given a Curtis-Tits amalgam over Fq with connected 3-spherical
diagram Γ there is a spanning tree Σ such that the set of edges in E Γ − E Σ =
{{is , js } : s = 1, 2, . . . , r} has the property that
(1) (G{is ,js} , gis ,js (Gis ), gis ,js (Gjs )) has type A2 (q es ), where es is some power
of 2.
(2) There is a loop Λs containing {is , js } such that any vertex group of Λs
l
is isomorphic to SL2 (q es 2 ) for some l ≥ 0.
Proof Induction on the rank r of H 1 (Γ, Z). If r = 0, then there is no loop at
all and we are done.
Consider the collection of all edges {i, j} of Γ such that Γ{i,j} has type A2
and H 1 (Γ − {i, j}, Z) has rank r − 1, and choose one such that Gi ∼
= SL2 (q e1 )
where e1 is minimal among all these edges. Next replace Γ by Γ−{i, j} and use
induction. Suppose {is , js } | s = 1, 2, . . . , r} is the resulting selection of edges
so that Σ = Γ − {is , js } | s = 1, 2, . . . , r} is a spanning tree and condition (1)
is satisﬁed. Note that by choice of these edges, also condition (2) is satisﬁed
by at least one of the loops of Γ − {{it , jt } : t = 1, 2, . . . , s − 1} that contains
{is , js }. Note that this uses the fact that by 3-sphericity every vertex belongs
to at least one subdiagram of type A2 .

Definition 4.21. Fix a connected 3-spherical diagram Γ and a prime power
q. Let Σ be a spanning tree and let the set of edges E Γ − E Σ = {{is , js } : s =
1, 2, . . . , r} together with the integers {es : s = 1, 2, . . . , r} satisfy the conclusions of Lemma 4.20. Let CT(Γ, q) be the collection of isomorphism classes of
Curtis-Tits amalgams of type Γ(q) and let G = {Gi , Gi,j , gi,j | i, j ∈ I} be the
standard Curtis-Tits amalgam over Fq with diagram Γ as in Subsection 4.4.
Consider the following map:
r
Y
κ:
Aut(Fqes ) × hτ i → CT(Γ).
s=1

κ((αs )rs=1 )

where
is the isomorphism class of the amalgam G + = G ((αs )rs=1 )
given by setting gj+s ,is = gj ,i ◦ αs for all s = 1, 2, . . . , r.
s s

CURTIS-TITS AND PHAN AMALGAMS

27

We now have
Corollary 4.22. The map κ is onto.
Proof Note that, for each s = 1, 2, . . . , r, the Curtis-Tits standard pair
(G{is ,js } , gis ,js (Gis ), gis ,js (Gjs )) has type A2 (q es ) and so Cjs = Aut(Fqes ).
Thus the claim is an immediate consequence of Proposition 4.16.

We note that if we select Σ diﬀerently, the map κ will still be onto. However,
the “minimal” choice made in Lemma 4.20 ensures that κ is injective as well,
as we will see.
Lemma 4.23. Suppose Γ(q) is a 3-spherical diagram Γ that is a simple loop.
Then, κ is injective.
φ

Proof Suppose there is an isomorphism κ(α) = G −→ G + = κ(β), for some
α, β ∈ Aut(Fq ) × hτ i. Write I = {0, 1, . . . , n − 1} so that {i, i + 1} ∈ E Γ
for all i ∈ I (subscripts modulo n). Without loss of generality assume that
(i1 , j1 ) = (1, 0) so that by Proposition 4.16 we may assume that gi,j = gi,j =
+
gi,j
for all (i, j) 6= (1, 0). This means that a : Ci,i+1 → Ci × Ci+1 sends φi,i+1
to (φi , φi+1 ) for any edge {i, i + 1} =
6 {0, 1}. Now note that by minimality of
q, Ci (and Ci,i+1 ) has a quotient Ci (and Ci,i+1) isomorphic to hAut(Fq )i ×
hτ i for every i ∈ I, by considering the action of Ci on the subgroup of Gi
isomorphic to SL2 (q). By Part 4 and 5 of Lemma 4.13 the maps a−1
i+1,i and
ai,i+1 induce isomorphisms Ci → Ci,i+1 and Ci,i+1 → Ci+1 , which compose to
an isomorphism
◦ gi,i+1 ◦ φi ◦ g−1
◦ gi+1,i ,
φi 7→ g−1
i+1,i
i,i+1
sending the image of τ and α in Ci to the image of τ (and α respectively) in
Ci+1 , where α : x 7→ xp for x in the appropriate extension of Fq deﬁning Gi,i+1 .
Concatenating these maps along the path {1, 2, . . . , n − 1, 0} and considering
the edge {0, 1} we see that the images of β −1φ1 α and φ1 in C1 = C1 coincide.
Since C1 is abelian this means that β = α.

Theorem 4.24. Let Γ be a connected 3-spherical diagram with spanning tree
Σ and set of edges E Γ − E Σ = {{is , js } : s = 1, 2, . . . , r} together with the
integers {es : s = 1, 2, . . . , r} satisfying theQconclusions of Lemma 4.20. Then
κ is a bijection between the elements of rs=1 Aut(Fqes ) × hτ i and the type
preserving isomorphism classes of Curtis-Tits amalgams with diagram Γ over
Fq .
Proof Again, it suﬃces to show that κ is injective. This in turn follows from
Lemma 4.23, for if two amalgams are isomorphic (via a type preserving isomorphism), then the amalgams induced on subgraphs of Γ must be isomorphic

28

BLOK, HOFFMAN, AND SHPECTOROV

and Lemma 4.23 shows that κ is injective on the subamalgams supported by
the loops Λs (s = 1, 2, . . . , r).


5. Classification of Phan amalgams
5.1. Introduction
The classiﬁcation problem is formulated as follows: Determine, up to isomorphism of amalgams, all Phan amalgams G with given diagram Γ possessing a
non-trivial (universal) completion.
5.2. Classification of Phan amalgams with 3-spherical diagram
5.2.1. Tori in Phan standard pairs. Let G = {Gi,j , Gi , gi,j | i, j ∈ I} be a
Phan amalgam over Fq with 3-spherical diagram Γ = (I, E). This means that
the subdiagram of Γ induced on any set of three vertices is spherical. This is
equivalent to Γ not containing triangles of any kind and such that no vertex
is on more than one C2 -edge.
Definition 5.1. For any i, j ∈ I with {i, j} ∈ E Γ, let
Dji = NGi,j (gj,i (Gj )) ∩ gi,j (Gi )
Lemma 5.2. Suppose that (G, G1 , G2 ) is a Phan standard pair of type Γ(q)
as in Subsection 2.3.
(1) If Γ(q) = A2 (q), then hD21 , D12 i is the standard torus stabilizing the
orthonormal basis {e1 , e2 , e3 }. Here D21 (resp. D12 ) is the stabilizer in
this torus of e1 (resp. e3 ).
(2) If Γ(q) = C2 (q), then hD21 , D12 i is the standard torus stabilizing the
basis {e1 , e2 , e3 = f1 , e4 = f2 } which is hyperbolic for the symplectic
form of Sp4 (q 2 ) and orthonormal for the unitary form of SU4 (q); Here
D12 (resp. D21 ) is the stabilizer of he2 i and hf2 i (resp. the pointwise
stabilizer of both he1 , f1 i and he2 , f2 i). Thus,
D12 = hdiag(1, a, 1, aσ ) : a ∈ Fq2 with aaσ = aq+1 = 1i,
D21 = hdiag(a, aσ , aσ , a) : a ∈ Fq2 with aaσ = aq+1 = 1i.
(3) In either case, for {i, j} = {1, 2}, Dji = CGi,j (Dij ) ∩ Gi and Dij is the
unique torus of Gj normalized by Dji .
Proof Parts 1. and 2. as well as the ﬁrst claim of Part 3. are straightforward
matrix calculations. As for the last claim note that in both cases, Dji acts
diagonally on Gj viewed as SU2 (q) in it natural representation V via the
standard identiﬁcation map; in fact (in the case C2 , D12 acts even innerly
on G1 ). If Dji normalizes a torus D′ in Gj then it will have to stabilize its
eigenspaces. Since q + 1 ≥ 3, the eigenspaces of Dji in its action on V have

CURTIS-TITS AND PHAN AMALGAMS

29

dimension 1, so Dji and D′ must share these eigenspaces. This means that
D′ = Dij .

5.2.2. Property (D) for Phan amalgams. We state Property (D) for 3-spherical
Phan amalgams, extending the deﬁnition from [4] which was given for CurtisTits amalgams under the assumption that q ≥ 4.
Definition 5.3. (property (D)) We say that G has property (D) if there is a
system of tori D = {Di : i ∈ I} such that for all edges {i, j} ∈ E Γ we have
gi,j (Di ) = Dji .
Lemma 5.4. Suppose that G has a completion (G, γ) so that γi is non-trivial
for all i ∈ I. Then, for any i, j, k ∈ I such that {i, j}, {j, k} ∈ E Γ, there is a
torus Dj ≤ Gj such that gj,i (Dj ) = Dij and gj,k (Dj ) = Dkj . In particular, G
has property (D).
Proof First note that in case q = 2, the conclusion of the lemma is trivially
true as, for all i ∈ I, Gi ∼
= S3 has a unique Phan torus.
We now consider the general case. For Γ(q) = A3 (q), this was proved by
Bennett and Shpectorov in [1] (see also [4]). For completeness we recall the
argument, which applies in this more general case as well. We shall prove that
γ(Dij ) = γ(Dkj )
and then let Dj ≤ Gj be such that γ(Dj ) = γ(Dij ) = γ(Dkj ). Note that since
γj is non-trivial, it now follows that gj,i (Dj ) = Dij and gj,k (Dj ) = Dkj .
Recall that for any subgroup H of a group in G we’ll write H = γ(H).
We show that Dji is normalized by Dkj and use Lemma 5.2 to conclude that
Dji = Djk . To that end we let h ∈ Dkj and prove that hDji h−1 = Dji . To achieve
this we show that hDji h−1 is normalized by Dij and again use Lemma 5.2. So
now let g ∈ Dij and note that since Γ is 3-spherical, {i, k} 6∈ E Γ so that g and
h commute. In addition note that by Lemma 5.2, gDji g −1 = Dji . Therefore we
have
ghDji h−1 g −1 = hgDji g −1 h−1 = hDji h−1 ,
as required.



5.2.3. The coefficient system of a Phan amalgam.
Definition 5.5. We now ﬁx a standard Phan amalgam G = {Gi , Gi,j , gi,j |
i, j ∈ I} over Fq with diagram Γ(q), where for every i, j ∈ I, gi,j is the standard
identiﬁcation map of Deﬁnition 2.6. Then, G has property (D) with system of
tori D = {Di : i ∈ I} as in Lemma 5.2.
If G is any other non-collapsing Phan amalgam over Fq with diagram Γ, then
since all tori of Gi are conjugate under Aut(Gi ), by adjusting the inclusion

30

BLOK, HOFFMAN, AND SHPECTOROV

maps gi,j we can replace G by an isomorphic amalgam whose system of tori is
exactly D.
From now on we assume that G , D = {Di : i ∈ I} and G are as in Deﬁnition 5.5
Definition 5.6. Suppose that G = {Gi , Gi,j , gi,j | i, j ∈ I} is a Phan amalgam
with connected 3-spherical diagram Γ having property (D). Let D = {Di : i ∈
I} be the associated system of tori. The coefficient system associated to G is
the collection A = {Ai , Ai,j , ai,j | i, j ∈ I} where, for any i, j ∈ I we set
Ai = NAut(Gi ) (Di ),
Ai,j = NAut(Gi,j ) (gi,j (Gi )) ∩ NAut(Gi,j ) (gj,i (Gj )),
−1
ai,j : Ai,j → Aj is given by restriction: ϕ 7→ gj,i
◦ ρi,j (ϕ) ◦ gj,i .

where ρi,j (ϕ) is the restriction of ϕ to Gj ≤ Gi,j .
From now on we let A be the coeﬃcient system associated to G with respect
to the system of tori D. The fact that the ai,j are well-deﬁned follows from
the following simple observation.
Lemma 5.7. For any i, j ∈ I with {i, j} ∈ E Γ, we have
Ai,j ≤ NAut(Gi,j ) (gi,j (Di )) ∩ NAut(Gi,j ) (gj,i (Dj )).
Proof The inclusion ≤ is immediate from the deﬁnitions.



The signiﬁcance for the classiﬁcation of Phan amalgams with the same system of tori is as follows:
Proposition 5.8. Suppose that G and G + are Phan amalgams of type G with
the same system of tori D = {Di : i ∈ I}.
+
(1) For all i, j ∈ I, we have gi,j = gi,j ◦ δi,j and gi,j = gi,j ◦ δi,j
for some
+
δi,j , δi,j ∈ Ai ,
(2) For any isomorphism φ : G → G + and i, j ∈ I, we have φi ∈ Ai ,
+
−1
φ{i,j} ∈ Ai,j , and ai,j (φ{i,j}) = δi,j
◦ φi ◦ δi,j
.
−1
Proof Part 1. follows since, for any i, j ∈ I we have gi,j
◦ gi,j ∈ Aut(Gi ) and

gi,j (Di ) = gi,j (Di ).
Part 2. follows from Lemma 5.7 since, for any i, j ∈ I,
+
+
(Gi,j , gi,j (Gi ), gj,i (Gj )) = (Gi,j , gi,j (Gi ), gj,i (Gj )) = (Gi,j , gi,j
(Gi ), gj,i
(Gj )).


We now determine the groups appearing in a coeﬃcient system by looking
at standard pairs.

CURTIS-TITS AND PHAN AMALGAMS

31

Lemma 5.9. Fix i ∈ I and let q be such that Gi ∼
= SU2 (q). Then,
A i = Ti ⋊ Ci ,
where Ti is the subgroup of diagonal automorphisms in PGU2 (q) and Ci =
Aut(Fq2 ).
Proof This follows from the fact that via the standard embedding map gi,j
the groups Di of the system of tori are the subgroups of standard diagonal
matrices in SU2 (q).
To see this note that Gi ∼
= PGU2 (q)⋊Aut(Fq2 ).
= SU2 (q) and that Aut(Gi ) ∼
Also, Di = hdi for some d = diag(ζ, ζ q ) and ζ a primitive q + 1-th root of 1 in
Fq2 . A quick calculation now shows that τ and σ are the same in their action,
which is inner and one veriﬁes that NGU2 (q) (Di ) = hτ, diag(a, b) : a, b ∈ F2q i. 
Lemma 5.10. Let A be the coefficient system associated to the standard Phan
amalgam G of type Γ(q) and the system of tori D.
If Γ = A1 × A1 , we have Gi,j = Gi × Gj , gi,j and gj,i are identity maps,
and
(5.1)
Ai,j = Ai × Aj ∼
= Ti,j ⋊ Ci,j .
where Ti,j = Ti × Tj and Ci,j = Ci × Cj . Otherwise,
Ai,j = Ti,j ⋊ Ci,j ,
where Ci,j = Aut(Fq2 ) and Ti,j denotes the image of the standard torus T in
Aut(Gi,j ). Note that T is as follows
hdiag(a, b, c) : a, b, c ∈ Fq2 with aaσ = bbσ = ccσ = 1i if Γ = A2 ,
hdiag(cσ b, ab, c, aσ ) : a, b, c ∈ Fq2 with aaσ = bbσ = ccσ = 1i if Γ = C2 .
Remark 5.11.
(1) In case Γ = C2 , G ∼
= Sp4 (q) is realized as Sp4 (q 2 ) ∩
SU4 (q) with respect to a basis that is hyperbolic for the symplectic
form and orthonormal for the unitary form, and Aut(Fq2 ) acts entrywise on these matrices. Moreover, τ acts as transpose-inverse on these
matrices.
(2) In all cases τ coincides with σ.
Proof The A1 × A1 case is self evident. Now consider the case Γ = A2 .
As in the proof of Lemma 5.9, Aut(Fq2 ) ≤ NAut(Gi,j ) (Gi ) ∩ NAut(Gi,j ) (Gj ),
Aτ = t A−1 = Aσ and Aut(Gi,j ) ∼
= PGU3 (q)⋊Aut(Fq2 ), so it suﬃces to consider
linear automorphisms. As before this is an uncomplicated calculation.
Now consider the case Γ = C2 . Writing ΓL(V ) ∼
= GL4 (q 2 ) ⋊ Aut(Fq2 ) with
respect to the basis E = {e1 , e2 , e3 = f1 , e4 = f2 }, which is hyperbolic for the
symplectic form of Sp4 (q 2 ) and orthonormal for the unitary form of SU4 (q),
we have Gi,j = Sp4 (q 2 ) ∩ SU4 (q).

32

BLOK, HOFFMAN, AND SHPECTOROV

There is an isomorphism Φ : Gi,j → Sp4 (q) as in [21]. Abstractly, we have
Aut(Sp4 (q)) = GSp4 (q) ⋊ Aut(Fq ) (with respect to a suitable basis E for V ).
Since the embedding of Sp4 (q) into Sp4 (q 2 ) is non-standard, we are reconstructing the automorphism group here.
We ﬁrst note that changing bases just replaces Aut(Fq2 ) with a diﬀerent
complement to the linear automorphism group. As for linear automorphisms
we claim that
GSp4 (q 2 ) ∩ GU4 (q) = GSp4 (q)
(viewing the latter as a matrix group w.r.t. E). Clearly, up to a center, we
have Gi,j ≤ GSp4 (q 2 )∩GU4 (q) ≤ GSp4 (q) and we note that GSp4 (q)/ Sp4 (q) ∼
=
∗ 2
∗
∗
(Fq ) /(Fq ). Thus for q even, the claim follows. For q odd, let Fq2 = hζi and
deﬁne β = diag(ζ q−1 , ζ q−1, 1, 1). Then β ∈ GSp4 (q 2 ) ∩ GU4 (q) acts on Gi,j as
diag(ζ q , ζ q , ζ, ζ), which scales the symplectic form of Sp4 (q 2 ) by ζ q+1. By [21]
the form of Sp4 (q) is proportional and since ζ q+1 is a non-square in Fq , β is
a linear outer automorphism of Sp4 (q). Thus, GSp4 (q) = hSp4 (q), βi and the
claim follows.
We now determine Ai,j . First we note that β, as well as the group Aut(Fq2 )
with respect to the basis E, clearly normalize Gi and Gj hence by Lemma 5.7,
Aut(Fq2 ) ≤ Ai,j . So it suﬃces to determine inner automorphisms of Sp4 (q)
normalizing Dji and Dij .
Any inner automorphism in Sp4 (q) is induced by an inner automorphism
of Sp4 (q 2 ). So now the claim reduces to a matrix calculation in the group
Sp4 (q 2 ).

Next we describe the restriction maps ai,j for Phan amalgams made up of
a single standard pair with trivial inclusion maps.
Lemma 5.12. Let A be the coefficient system of the standard Phan amalgam G over Fq with diagram Γ and system of tori D. Fix i, j ∈ I and
let (Gi,j , Gi , Gj ) be a Phan standard pair in G with diagram Γi,j . Denote
a = (aj,i , ai,j ) : Ai,j → Ai × Aj . Then, we have the following:
(1) If Γi,j = A1 × A1 , then a is an isomorphism inducing Ti,j ∼
= Ti × Tj
∼
and Ci,j = Ci × Cj .
(2) If Γi,j = A2 or C2 , then a induces an isomorphism Ti,j → Ti × Tj .
(3) If Γ(q) = A2 (q) or Γ(q) = C2 (q), then a : Ci,j → Ci × Cj is given by
α 7→ (α, α) (for α ∈ Aut(Fq2 )) which is a diagonal embedding.
Proof 1. This is immediate from Lemma 5.10.
For the remaining cases, recall that for any ϕ ∈ Ai,j , we have ai,j : ϕ 7→
−1
gj,i ◦ ρi,j (ϕ) ◦ gj,i , where ρi,j (ϕ) is the restriction of ϕ to Gj ≤ Gi,j (Deﬁnition 5.6) and gi,j is the standard identiﬁcation map of Deﬁnition 2.6. Note

CURTIS-TITS AND PHAN AMALGAMS

33

that the standard identiﬁcation map transforms the automorphism ρj,i (ϕ) of
Gi essentially to the “same” automorphism ϕ of Gi .
First let Γ(q) = A2 (q). The map a is well-deﬁned. On T, it is induced by
the homomorphism
diag(ac, c, ec) 7→ (diag(1, e), diag(a, 1)),
where a, c, e ∈ Fq2 are such that aaσ = ccσ = eeσ = 1. Note that the kernel is
Z(T) so that a is injective. The map is obviously surjective, so we are done.
Thus if we factor a by Ti,j and Ti × Tj , we get
(5.2)

Ci,j ֒→Ci × Cj

which is a diagonal embedding given by αr 7→ (αr , αr ), where r ∈ N and
α : x 7→ xp for x ∈ Fq2 .
Next let Γ(q) = C2 (q). We can rewrite the elements of T as a diagonal
matrix diag(xyz, xz, zy −1 , z), by taking z = aσ , y = (ac)−1 , x = a2 b. On T
the map a is induced by the homomorphism
diag(xyz, xz, y −1 z, z) 7→ (diag(y, 1), diag(x, 1))
with kernel {diag(z, z, z, z) : z ∈ Fq2 with zz σ = 1} = Z(GU4 (q)). Clearly
a : Ti,j → Ti × Tj is an isomorphism. Taking the quotient over these groups,
a induces a diagonal embedding as in (5.2), where we now interpret it in the
C2 (q) setting.

5.2.4. A standard form for Phan amalgams. Suppose that G = {Gi , Gi,j , gi,j |
i, j ∈ I} is a Phan amalgam over Fq with 3-spherical diagram Γ. Without loss
of generality we will assume that all inclusion maps gi,j are the standard identiﬁcation maps of Deﬁnition 2.6. By Lemma 5.4, G has Property (D) and possesses a system D = {Di : i ∈ I} of tori, which, as noted in Deﬁnition 5.5 via
the standard embeddings gi,j can be identiﬁed with those given in Lemma 5.2.
We wish to classify all Phan amalgams G = {Gi , Gi,j , gi,j | i, j ∈ I} over
Fq with the same diagram as G . As noted in Deﬁnition 5.5 we may assume
that all such amalgams share D. Let A = {Ai , Ai,j , ai,j | i, j ∈ I} be the
coeﬃcient system of G associated to D. By Proposition 5.8, we may restrict
to those amalgams whose connecting maps are of the form gi,j = gi,j ◦ δi,j for
δi,j ∈ Ai for all i ∈ I.
Definition 5.13. The trivial support of G (with respect to G ) is the set
{(i, j) ∈ I × I | gi,j = gi,j } (that is, δi,j = idGi in the notation of Proposition 5.8). The word “trivial” derives from the assumption that the gi,j ’s are
the standard identiﬁcation maps of Deﬁnition 2.6.
Fix some spanning tree Σ ⊆ Γ and suppose that E − E Σ = {{is , js } : s =
1, 2, . . . , r} so that H1 (Γ, Z) ∼
= Zr . We now have

34

BLOK, HOFFMAN, AND SHPECTOROV

Proposition 5.14. There is a Phan amalgam G (Σ) with the same diagram as
G and the same D, which is isomorphic to G and has the following properties:
(1) G has trivial support S = {(i, j) ∈ I × I | {i, j} ∈ E Σ} ∪ {(is , js ) : s =
1, 2 . . . , r}.
(2) for each s = 1, 2, . . . , r, we have gjs ,is = gj ,i ◦ γjs ,is , where γjs,is ∈ Cjs .
s s

Lemma 5.15. There is a Phan amalgam G + over Fq with the same diagram as
G and the same D, which is isomorphic to G and has the following properties:
For any u, v ∈ I, if gu,v = gu,v ◦ γu,v ◦ du,v , for some γu,v ∈ Cu and du,v ∈ Tu ,
+
then gu,v
= gu,v ◦ γu,v .
Proof The proof follows the same steps as that of Lemma 4.17 using Part 2
of Lemma 5.12 instead of Lemma 4.13 Part 2.

By Lemma 5.15 in order to prove Proposition 5.14 we may now assume that
gu,v = gu,v ◦ γu,v for some γu,v ∈ Cu for all u, v ∈ I.
We now prove a Corollary for Phan amalgams analogous to, but stronger
than Corollary 4.18. To this end consider the situation of Figure 1 interpreted
in the Phan setting.
Corollary 5.16. With the notation introduced in Figure 1, fix the maps γi,j ,
+
+
γi,j
, φi ∈ Ci as well as γj,i ∈ Cj . Then for any one of γj,i
, φj ∈ Cj , there
exists a unique choice γ ∈ Ci for the remaining map in Cj so that there exists
φi,j making the diagram in Figure 1 commute.
Proof This follows immediately from the fact that the maps aj,i : Ci,j → Ci
and ai,j : Ci,j → Cj in part 3. of Lemma 5.12 are isomorphisms.

Proof (of Proposition 5.14) The proof follows the same steps as that of Proposition 4.16, replacing Lemma 4.17 and Corollary 4.18 by Lemma 5.15 and
Corollary 5.16.

5.2.5. Classification of Phan amalgams with 3-spherical diagram. In the case
where G is a Phan amalgam over Fq whose diagram is a 3-spherical tree,
Proposition 5.14 says that G ∼
= G.
Theorem 5.17. Suppose that G is a Curtis-Tits amalgam with a diagram that
is a 3-spherical tree. Then, G is unique up to isomorphism. In particular any
Phan amalgam with spherical diagram is unique.
Definition 5.18. Fix a connected 3-spherical diagram Γ and a prime power
q. Let Σ be a spanning tree and let the set of edges E Γ − E Σ = {{is , js } : s =
1, 2, . . . , r} together with the integers {es : s = 1, 2, . . . , r} satisfy the conclusions of Lemma 4.20. Note that since in the Phan case we do not have
subdiagrams of type 2A3 (q), we have es = 1 for all s = {1, 2, . . . , r}.

CURTIS-TITS AND PHAN AMALGAMS

35

Let Ph(Γ, q) be the collection of isomorphism classes of Phan amalgams of
type Γ(q) and let G = {Gi , Gi,j , gi,j | i, j ∈ I} be a Phan amalgam over Fq
with diagram Γ.
Consider the following map:
r
Y
κ:
Aut(Fq2 ) → Ph(Γ).
s=1

κ((αs )rs=1 )

where
is the isomorphism class of the amalgam G + = G ((αs )rs=1 )
given by setting gj+s ,is = gjs ,is ◦ αs for all s = 1, 2, . . . , r.
As for Curtis-Tits amalgams, one shows the following.
Corollary 5.19. The map κ is onto.
Lemma 5.20. Suppose Γ(q) is a 3-spherical diagram Γ that is a simple loop.
Then, κ is injective.
Proof The proof is identical to that of Lemma 4.23 replacing Proposition 4.16
by Proposition 5.14 and Lemma 4.13 by Lemma 5.12, and noting that in the
Phan case, we can consider the group Ci and Ci,j themselves rather than some
suitably chosen quotient.

Theorem 5.21. Let Γ be a connected 3-spherical diagram with spanning tree
Σ and set of edges E Γ −E
QΣ = {{is , js } : s = 1, 2, . . . , r}. Then κ is a bijection
between the elements of rs=1 Aut(Fq2 ) and the isomorphism classes of CurtisTits amalgams with diagram Γ over Fq .

Proof This follows from Lemma 5.20 just as Theorem 4.24 follows from
Lemma 4.23.


References
[1] C. D. Bennett and S. Shpectorov. A new proof of a theorem of Phan. J. Group Theory,
7(3):287–310, 2004.
[2] R. J. Blok and C. G. Hoffman. 1-cohomology of simplicial amalgams of groups. J.
Algebraic Combin., 37(2):381–400, 2013.
[3] R. J. Blok and C. G. Hoffman. Curtis–Tits groups generalizing Kac–Moody groups of
type Ãn−1 . J. Algebra, 399:978–1012, 2014.
[4] R. J. Blok and C. Hoffman. A classfication of Curtis-Tits amalgams. In N. Sastry,
editor, Groups of Exceptional Type, Coxeter Groups and Related Geometries, volume
149 of Springer Proceedings in Mathematics & Statistics, pages 1–26. Springer, January
2014.
[5] R. J. Blok and C. G. Hoffman. Curtis-Tits groups of simply-laced type. To appear in
J. Comb. Th. Ser. A.
[6] P. E. Caprace. On 2-spherical Kac-Moody groups and their central extensions. Forum
Math., 19(5):763–781, 2007.
[7] C. W. Curtis. Central extensions of groups of Lie type. J. Reine Angew. Math., 220:174–
185, 1965.

36

BLOK, HOFFMAN, AND SHPECTOROV

[8] J. Dunlap. Uniqueness of Curtis-Phan-Tits amalgams. PhD thesis, Bowling Green State
University, 2005.
[9] D. Gorenstein. The classification of finite simple groups. Vol. 1. The University Series
in Mathematics. Plenum Press, New York, 1983. Groups of noncharacteristic 2 type.
[10] D. Gorenstein, R. Lyons, and R. Solomon. The classification of the finite simple groups.
Number 2. Part I. Chapter G, volume 40 of Mathematical Surveys and Monographs.
American Mathematical Society, Providence, RI, 1996. General group theory.
[11] D. Gorenstein, R. Lyons, and R. Solomon. The classification of the finite simple groups.
Number 3. Part I. Chapter A, volume 40 of Mathematical Surveys and Monographs.
American Mathematical Society, Providence, RI, 1998. Almost simple K-groups.
[12] D. Gorenstein, R. Lyons, and R. Solomon. The classification of the finite simple groups.
Number 4. Part II. Chapters 1–4, volume 40 of Mathematical Surveys and Monographs.
American Mathematical Society, Providence, RI, 1999. Uniqueness theorems, With
errata: ıt The classification of the finite simple groups. Number 3. Part I. Chapter A
[Amer. Math. Soc., Providence, RI, 1998; MR1490581 (98j:20011)].
[13] D. Gorenstein, R. Lyons, and R. Solomon. The classification of the finite simple groups.
Number 5. Part III. Chapters 1–6, volume 40 of Mathematical Surveys and Monographs.
American Mathematical Society, Providence, RI, 2002. The generic case, stages 1–3a.
[14] D. Gorenstein, R. Lyons, and R. Solomon. The classification of the finite simple groups.
Number 6. Part IV, volume 40 of Mathematical Surveys and Monographs. American
Mathematical Society, Providence, RI, 2005. The special odd case.
[15] R. Gramlich. Weak Phan systems of type Cn . J. Algebra, 280(1):1–19, 2004.
[16] F. G. Timmesfeld. Presentations for certain Chevalley groups. Geom. Dedicata,
73(1):85–117, 1998.
[17] F. G. Timmesfeld. On the Steinberg-presentation for Lie-type groups. Forum Math.,
15(5):645–663, 2003.
[18] F. G. Timmesfeld. The Curtis-Tits-presentation. Adv. Math., 189(1):38–67, 2004.
[19] F. G. Timmesfeld. Steinberg-type presentation for Lie-type groups. J. Algebra,
300(2):806–819, 2006.
[20] R. Gramlich. Weak Phan systems of type Cn . J. Algebra, 280(1):1–19, 2004.
[21] R. Gramlich, C. Hoffman, and S. Shpectorov. A Phan-type theorem for Sp(2n, q). J.
Algebra, 264(2):358–384, 2003.
[22] R. Gramlich, M. Horn, and W. Nickel. The complete Phan-type theorem for Sp(2n, q).
J. Group Theory, 9(5):603–626, 2006.
[23] R. Gramlich, M. Horn, and W. Nickel. The complete Phan-type theorem for Sp(2n, q).
J. Group Theory, 9(5):603–626, 2006.
[24] B. Mühlherr. Locally split and locally finite twin buildings of 2-spherical type. J. Reine
Angew. Math., 511:119–143, 1999.
[25] K.-W. Phan. A characterization of the unitary groups PSU(4, q 2 ), q odd. J. Algebra,
17:132–148, 1971.
[26] K. W. Phan. On groups generated by three-dimensional special unitary groups. I. J.
Austral. Math. Soc. Ser. A, 23(1):67–77, 1977.
[27] K.-W. Phan. On groups generated by three-dimensional special unitary groups. II. J.
Austral. Math. Soc. Ser. A, 23(2):129–146, 1977.
[28] O. Schreier and B. Van der Waerden. Die automorphismen der projektiven gruppen.
Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 6(1):303–
322, December 1928.

CURTIS-TITS AND PHAN AMALGAMS

37

[29] J. Tits. Twin buildings and groups of Kac-Moody type. In Groups, combinatorics &
geometry (Durham, 1990), volume 165 of London Math. Soc. Lecture Note Ser., pages
249–286. Cambridge Univ. Press, Cambridge, 1992.
[30] R. A. Wilson. The finite simple groups, volume 251 of Graduate Texts in Mathematics.
Springer-Verlag London Ltd., London, 2009.
Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, oh 43403, U.S.A.
Current address: School of Mathematics, University of Birmingham, Edgbaston, B15
2TT, U.K.
E-mail address: blokr@member.ams.org
School of Mathematics, University of Birmingham, Edgbaston, B15 2TT,
U.K.
E-mail address: C.G.Hoffman@bham.ac.uk
School of Mathematics, University of Birmingham, Edgbaston, B15 2TT,
U.K.
E-mail address: S.V.Shpectorov@bham.ac.uk

