APPLICATIONS OF THE ALEXANDER IDEALS TO THE ISOMORPHISM PROBLEM FOR FAMILIES OF GROUPS

APPLICATIONS OF THE ALEXANDER IDEALS TO THE
ISOMORPHISM PROBLEM FOR FAMILIES OF GROUPS
DO VIET HUNG AND VU THE KHOI
Abstract. In this paper, we use the Alexander ideals of groups to solve the
isomorphism problem for the Baumslag-Solitar groups and a family of parafree
groups introduced by Baumslag and Cleary.

1. Introduction
1.1. Problems and results. The isomorphism problem is a fundamental problem
in group theory in which we have to decide whether two finitely presented groups
are isomorphic. In the most general form, the isomorphism problem was proved
to be unsolvable. Therefore it makes sense to restrict the problem to a special
class of groups. The isomorphism problem for certain families of groups has been
considered by many authors. For those works which are particularly related to our,
see [7, 9, 12, 14]. The purpose of this paper is to use the Alexander ideal, an algebraic
invariant of groups which was originated from topology, to study the isomorphism
problem for families of groups. In many cases, by computing the Alexander ideals
of the groups in the family, we can deduce that two groups in that family are not
isomorphic. Our main results are the solutions of the isomorphism problems for the
Baumslag-Solitar groups and a family of parafree groups.
The last part of this section is devoted to reviewing backgrounds on Alexander
modules and ideals of a finitely presented group. In section two, we present a proof
of a result of Moldavanskii [14] on the isomorphism problem of the Baumslag-Solitar
2010 Mathematics Subject Classification. Primary 20Exx, Secondary 57M05.
Key words and phrases. Group isomorphism problem, Alexander ideal, parafree groups.
This research is funded by Vietnam National Foundation for Science and Technology Developmen(NAFOSTED) under grant number 101.01-2011.46.
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DO VIET HUNG AND VU THE KHOI

groups. Section three solves the isomorphism problem for a family of parafree groups
Ki,j which was introduced by Baumslag and Cleary in [6]. In particularly, by using
the Alexander ideal, we show that all the groups in that family are distinct.
Acknowledgments. The authors would like to express their gratitude to the
Vietnam Institute for Advanced Study in Mathematics for support during the writing of the paper.
1.2. Alexander ideals. In this subsection, we will present some backgrounds on
the Alexander ideals of finitely presented groups. For more details, the readers are
referred to [1, 11, 13, 15].
Let G = x1 , . . . , xk |r1 , . . . , rl be a finitely presented group and
ab(G) := H1 (G; Z)/(torsion)
be its maximal free abelian quotient. Suppose that (X, p) be a pointed CW- complex
˜ → X be the covering corresponding to φ : G →
such that π1 (X, p) = G. Let π : X
˜ p˜; Z) the relative homology. Then the
ab(G) and p˜ := π −1 (p). We denote by H1 (X,
˜ makes H1 (X,
˜ p˜; Z) an Z[ab(G)]−module,
desk transformation action of ab(G) on X
which is called the Alexander module of G. We can also describe the Alexander
module in a purely algebraic way as (see [13])
M = m(G)/m(ker φ) · m(G).
Where M(H), for a subgroup H ⊂ G, is the augmentation ideal of Z[G] generated
by (h − 1) : h ∈ H .
Suppose that we fix an isomorphism χ : ab(G) → Zn , then the group ring
±1
±1
Λ := Z[ab(G)] can be identified with Z[t±1
1 , t2 , . . . , tn ]. It is well-known that the

˜ p˜; Z) is a finitely generated Λ-module, so we choose
Alexander module M = H1 (X,

a presentation:
A

→ Λl → Λk → M.

(1)

Let A be the presentation matrix of the Alexander module as above. The ith Alexander
ideal of G is the ideal generated by all the (k − i) × (k − i) minors of the presentation
matrix A.


APPLICATIONS OF THE ALEXANDER IDEALS TO THE ISOMORPHISM PROBLEM

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The Alexander ideals do not depend on the choices of the CW-complex X and
the presentation (1). However we have a freedom in choosing the isomorphism χ
above, so the Alexander ideals are invariants of the group G up to a monomial
automorphism of Λ. that is an automorphism such that ϕ(ti ) = t1ai1 t2ai2 . . . tanin , i =
1, . . . , n, where (aij ) is a matrix belonging to GL(n, Z).
There is an effective algorithm to compute the Alexander modules and ideals by
using Fox’s free differential calculus [10] which we will describe briefly below.
Suppose that Fk = x1 , . . . , xk | is the free group on k generators. Let ǫ : ZFk → Z
be the augmentation homomorphism defined by ǫ(

ni gi ) =

ni . The j th partial

Fox derivative is a linear operator ∂j : ZFk → ZFk which is uniquely determined by
the following rules:
∂j (1) = 0; ∂j (xi ) = δij ;

∂j (uv) = ∂j (u)ǫ(v) + u∂j (v).
As consequences of the above rules we get:
i) ∂i (xni ) = 1 + xi + x2i + · · · + xin−1 for all n ≥ 1.
−1
−2
−n
ii) ∂i (x−n
for all n ≥ 1.
i ) = −xi − xi − · · · − xi

Let G = x1 , . . . , xk |r1 , . . . , rl be a group as above. From the quotient map
φ

Fk → G → ab(G) we get a map Φ : ZFk → Λ.
The results of Fox’s free differential calculus say that the Jacobian matrix
J := (∂j ri )Φ : Λl → Λk
is a presentation matrix for the Alexander module of G. Thus, we have an effective
method to find the Alexander ideals.
2. Isomorphism problem for the Baumslag-Solitar Groups
The Baumslag-Solitar group B(m, n), where m, n are non-zero integers, can be
defined by:
B(m, n) = a, b a−1 bm a = bn = a, b a−1 bm ab−n = 1 .

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DO VIET HUNG AND VU THE KHOI

The Baumslag-Solitar groups are first given by Baumslag and Solitar [8] as an example of one-relator non-Hopfian groups. The Baumslag-Solitar groups attract attention of many authors since they serve as a rich source of examples and counterexamples for many questions in group theory.
The isomorphism problem for Baumslag-Solitar groups was solved by Moldavanskii [14] in 1991. As a first illustration of our method, we will show that the solution
of the isomorphism problem for the Baumslag-Solitar groups can be deduced quickly
from the computation of Alexander ideals.
Proposition 2.1. The first Alexander ideal of the group B(m, n) is
±1
i) I = ((1 + t2 + ... + t2n−1 )(1 − t1 ), 1 − tn2 ) ⊆ Z[t±1
1 , t2 ] in the case m = n > 0;
±1 ±1
ii) I = ((1 + t2 + ... + t2−n−1 )(1 − t1 ), 1 − t−n
2 ) ⊆ Z[t1 , t2 ] in the case m = n < 0;

iii) I = (m − nt1 ) ⊆ Z[t1 ] in the case m = n.
Proof. To shorten the computations, we note without proof the following fact:
B(m, n) ∼
= B(−n, −m).
= B(−m, −n) ∼
= B(n, m) ∼
i) If m = n > 0, we can deduce that
ab(B(n, n)) ∼
= Zt1 ⊕ Zt2
a

→

t1

b

→

t2 .

±1
So the group ring Λ = Z[ab(B(n, n)] ∼
= Z[t±1
1 , t2 ].

Applying Fox’s free differential calculus to the relation r = a−1 bn ab−n , we have:
∂r
= −a−1 + a−1 bn ,
∂a
∂r
= a−1 (1 + b + b2 + ... + bn−1 ) − a−1 bn a(b−1 + ... + b−n ).
∂b
Therefore
Φ(

∂r
∂r
n
) = t−1
) = (1 + t2 + ... + t2n−1 )(t−1
1 (t2 − 1), Φ(
1 − 1).

∂a
∂b

We see that the first Alexander ideal of B(n, n) is I = ((1 + t2 + ... + t2n−1 )(1 −
t1 ), 1 − tn2 ).


APPLICATIONS OF THE ALEXANDER IDEALS TO THE ISOMORPHISM PROBLEM

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ii) The case m = n < 0, by using the fact that B(n, n) ∼
= B(−n, −n) and the
result of i), we find that the Alexander ideal is
I = ((1 + t2 + ... + t2−n−1 )(1 − t1 ), 1 − t−n
2 ).
iii) We will divide this case into several subcases:
a) m = n and m, n > 0. We see that
ab(B(m, n)) ∼
= Zt1
a

→

t1

b

→

0

We get Λ = Z[ab(B(m, n))] ∼
= Z[t±1
1 ].
Applying Fox’s free differential calculus to r = a−1 bm ab−n , we have:
∂r
= −a−1 + a−1 bm
∂a
∂r
= a−1 (1 + b + b2 + ... + bm−1 ) − a−1 bm a(b−1 + ... + b−n ).
∂b
From this computation we get
Φ(

∂r
∂r
) = 0, Φ( ) = mt−1
1 − n.
∂a
∂b

So the first Alexander ideal is I = (m − nt1 ).
b) m = n and m, n < 0. Since B(m, n) ∼
= B(−m, −n), it follows from the subcase
a) that the first Alexander ideal is I = (−m + nt) = (m − nt).
c) m > 0 and n < 0. This case is almost identical the subcase a) except a small
change :
∂r
= a−1 (1 + b + b2 + ... + bm−1 ) + a−1 bm a(1 + b + ... + b−n−1 ).

∂b
However, straightforward computations show that the first Alexander ideal is still
the same I = (m − nt1 ).
d) m < 0 and n > 0. We use the fact that B(m, n) ∼
= B(−m, −n) to reduce the
computation to the subcase c). So I = (−m + nt1 ) = (m − nt1 ).


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DO VIET HUNG AND VU THE KHOI

Theorem 2.2. (Moldavanskii [14]) The group B(m, n) and B(p, q) are isomorphic
if and only if for a suitable ǫ = ±1 either m = pǫ and n = qǫ or m = qǫ and n = pǫ.
Proof. We only prove the necessity since the sufficiency is obvious as we already
notice above. Now suppose that B(m, n) ∼
= ab(B(p, q))
= B(p, q). As ab(B(m, n)) ∼
we see that m = n implies p = q. So we only need to prove Theorem in the following
cases
i) The case m = n and p = q. We denote by
V (I) = {(t1 , t2 ) ∈ (C∗ )2 | f (t1 , t2 ) = 0 ∀ f ∈ I}
the zero locus of the first Alexander ideal I. By the results of Proposition 2.1 part
i) and ii), for the group B(n, n), V (I) consists of (|n| − 1) lines {t2 = e

2πik
|n|

}, k =

1, . . . |n| − 1, and an isolated point {t1 = 1, t2 = 1}. In particular, V (I) has |n| connected components. Therefore B(n, n) ∼
= B(q, q) implies |n| = |q|. So the Theorem
is proved in this case.
ii) The case m = n and p = q. Using Proposition 2.1, we get the equality of
ideals (m − nt1 ) = (p − qt1 ) in Z[t1 ] up to a monomial automorphism. In this case,
the monomial automorphism is just changing t ↔ t−1 . So this implies that either
m − nt1 = ±(p − qt1 ) or m − nt1 = ±(pt − q1 ). We get the required conclusion.
3. Isomorphism problem for a family of parafree groups
Parafree group is a class of groups which shares many similar properties with free
groups. However they may not necessarily be free. To give the precise definition,
we will need some notations. For a group G, let
G = γ1 (G) ≥ γ2 (G) ≥ · · · ≥ γn (G) ≥ · · ·
be the lower central series which is defined inductively by γn+1(G) := [G, γn (G)]. A
group is called residually nilpotent if ∩∞
i=1 γi (G) = 1.
A group G is called parafree if it is residually nilpotent and has the same nilpotent quotient as a given free group, that is, there exists a free group F such that


APPLICATIONS OF THE ALEXANDER IDEALS TO THE ISOMORPHISM PROBLEM

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G/γk (G) ∼
= F/γk (F ) for all k. Parafree group is studied for the first time by Baumslag [2]. Since then, several explicit families of parafree groups have been introduced,
see [3, 4, 5].
As the parafree groups enjoys many common properties with free groups, the
isomorphism problem for parafree groups is very hard. There have been some partial
results in the isomorphism problem for groups in the families of parafree groups
mention above. In [9] the authors study the isomorphism problem for the family
of parafree groups Gi,j := a, b, t| a−1 = [bi , a][bj , t] introduced by Baumslag in [4],

here [x, y] := x−1 y −1xy. In particular, they show that Gi,1 ∼
= G1,1 for i > 1 and
Gi,1 ∼
= Gj,1 for distinct primes i, j. So far, no other theoretical attack has been
carried out. Computational approach to distinguish parafree groups by enumerating
homomorphism to a fixed finite group has been done by some authors, see [7, 12].
However, this method can only distinguish finitely many groups in the family.
In the following, we use the Alexander ideals to distinguish parafree groups in the
family Ki,j which was introduced by Baumslag and Cleary in [6]:
Ki,j := a, s, t| ai [s, a] = tj , where i > 0, j > 1 are relatively prime.
We compute the second Alexander ideals of the groups in this family in the following
proposition.
Proposition 3.1. The second Alexander ideal of the group Ki,j above are given by
(i−1)j

I = (1 − tj2 , (1 + tj2 + . . . + t2

(i−1)j −1
t1

) − t2

(i−1)j

+ t2

Proof. Note that
ab(Ki,j ) ∼
= Zt1 ⊕ Zt2
s

→

t1

a

→

tj2 .

t

→

ti2 .

(j−1)i

, 1 + ti2 + . . . + t2

±1
We deduce that the group ring Λ = Z[ab(Ki,j )] ∼
= Z[t±1
1 , t2 ].

).


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DO VIET HUNG AND VU THE KHOI

Applying Fox’s free differential calculus to the relation r = ai s−1 a−1 sat−j , we
have:

∂r
= −ai s−1 + ai s−1 a−1 ,
∂s
∂r
= (1 + a + . . . + ai−1 ) − ai s−1 a−1 + ai s−1 a−1 s,
∂a
∂r
= −ai s−1 a−1 sa(t−1 + ... + t−j ).
∂t

So, we find that
Φ(

∂r
(i−1)j −1
−1
) = −tij
t1 ,
2 t1 + t2
∂s

∂r
(i−1)j
(i−1)j −1

(i−1)j
) = (1 + tj2 + . . . + t2
) − t2
t1 + t2
,
∂a
∂r
(j−1)i
).
Φ( ) = −(1 + ti2 + . . . + t2
∂b
From that, the proposition follows.
Φ(

Now we can solve the isomorphism problem for the family Ki,j .
Theorem 3.2. For i > 0, j > 1 relatively prime, all the group Ki,j in the family are
distinct.
Proof. We first find the zero set V (I) of the second Alexander ideal of Ki,j . It follows
from Proposition 3.1 that V (I) is the solution set of the system:



= 0 (1)
1 − tj2



(i−1)j
(i−1)j −1
(i−1)j

(1 + tj2 + . . . + t2
) − t2
t1 + t2
= 0 (2)




1 + ti + . . . + t(j−1)i
= 0 (3)
2
2

Using the fact that i, j are relatively prime, we get:



tj = 1


2
2πik
(1) + (3) ⇐⇒ tij
= 1 ⇐⇒ t2 = e j , k = 1, · · · , j − 1.
2



i


t2 = 1

Combining with equation (2), we find that V (I) consists of j − 1 points
{(t1 , t2 ) = (

2πik
1
, e j )}k=1,··· ,j−1 .
i+1


APPLICATIONS OF THE ALEXANDER IDEALS TO THE ISOMORPHISM PROBLEM

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Suppose that Ki,j ∼
= Km,n . By counting the cardinality of V (I), we deduce that
j = n. Now, suppose that
ϕ : C∗ × C∗ → C∗ × C∗
t1

→

ta1 tb2

t2

→

tc1 td2 .

be the monomial automorphism that maps the zero set of the second Alexander
ideal of Ki,j to that of Km,n . That is,
(ϕ(t1 ), ϕ(t2 )) ∈ {(

2πik
1
, e n )}k=1,··· ,n−1
m+1

∀(t1 , t2 ) ∈ {(

2πik
1
, e j )}k=1,··· ,j−1.
i+1

1 c
From that, we get |ϕ(t2 )| = ( i+1
) = 1 and therefore c = 0. Because the matrix

a b

belongs to GL(2, Z), from c = 0, we find that a = ±1. So we must have:
c d
1 ±1
1
|ϕ(t1 )| = ( i+1
) = m+1
. It follows that a = 1 and i = m.

In conclusion, we find that if Ki,j ∼
= Km,n then i = m and j = n.
Remark. For other families of parafree groups which were investigated in [7],
the Alexander ideals are found to be trivial. So the method described here does not
work.
References
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(1928), 275-306.
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The groups. Trans. Amer. Math. Soc 129 (1967), 308-321.
[3] Baumslag, Gilbert. Some groups that are just about free. Bull. Amer. Math. Soc. 73 (1967),
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[4] Baumslag, Gilbert. Groups with the same lower central sequence as a relatively free group.
II. Properties. Trans. Amer. Math. Soc 142 (1969), 507-538.
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Amer. Math. Soc., Providence, RI, 1994.
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no. 2, 191-201.


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[7] Baumslag, Gilbert, Cleary, Sean and Havas, George. Experimenting with infinite groups, I.
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[10] Fox. R. H. Free differential calculus I. Annals of Math. 57 (1953), 547560.
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47 (1997), no. 2, 555583.
[12] Lewis, Robert H., and Sal Liriano. Isomorphism classes and derived series of certain almostfree groups. Experiment. Math. 3 (1994), no. 3, 255-258.
[13] McMullen, Curtis T. The Alexander polynomial of a 3-manifold and the Thurston norm on
cohomology. Ann. Sci. cole Norm. Sup. (4) 35 (2002), no. 2, 153171.
[14] Moldavanskii, D. I. On the isomorphism of Baumslag-Solitar groups. Ukrainian Math. J. 43
(1991), no. 12, 15691571
[15] Suciu, A., Fundamental groups, Alexander invariants, and cohomology jumping loci. Topology
of algebraic varieties and singularities, 179223, Contemp. Math., 538, Amer. Math. Soc.,
Providence, RI, 2011.
Do Viet Hung, Ha Giang College of Education, Ha Giang, Viet Nam
E-mail address:
Vu The Khoi (Corresponding author), Institute of Mathematics, 18 Hoang Quoc
Viet, 10307, Hanoi, Vietnam
E-mail address: