Algebra and Logic, Vol. 48, No. 3, 2009
PREGROUPS AND THE BIG
POWERS CONDITION
A. V. Kvaschuk, A. G. Myasnikov,
1
and D. E. Serbin
1
UDC 512.5
Dedicated to V. N. Remeslennikov
on the occasion of his 70th birthday
Keywords: group, pregroup, universal group, big powers property.
We study groups having the big powers property BP. It is proved that if a pregroup satisfies some
natural axioms, then its universal group has this property. In particular, fundamental groups of
some graphs of groups have the big powers property if BP holds for edge and vertex subgroups
and a number of natural conditions are satisfied. The results obtained are applied to Lyndon’s
completions U(P )
Z[t]
of the universal group U(P ) with P satisfying the conditions mentioned.
1. PRELIMINARIES
The big powers condition has been used in various forms in different areas of group theory. Usually
it stays behind the scenes, shadowed by various geometric properties like hyperbolicity, toral relative hy-
perbolicity, or the classical cancelation argument. However, a detailed analysis shows that quite often an
argument does not require the full strength of geometric properties, but is based mostly on their pure
combinatorial surrogates such as CSA (Sec. 6) and BP (Sec. 2). Therefore, we may introduce an abstract
notion of the big powers condition in groups, which clarifies and streamlines proofs in many situations and
makes the argument much wider applicable. In this paper we lay down basics of a combinatorial approach
to groups with the BP property and point out some applications well beyond the classes of hyperbolic
or toral relatively hyperbolic groups. Furthermore, the BP condition is interesting in its own right, as a
non-Abelian analog of independence in a linear space.
Let G be a group and u =(u
1
,...,u
k
) a sequence of nontrivial elements of G.Wesaythatu
(1) is generic if neighbors in u do not commute: namely,
[u
i
,u
i+1
]
=1 forevery i ∈ [1,k− 1],
and
(2) is independent if there exists n = n(u) ∈ N such that for any α
1
,...,α
k

n,
u
α
1
1
...u
α
k
k

=1.
1
Department of Mathematics and Statistics, McGill University, Montreal, Canada; alexeim@math.mcgill.ca. Trans-
lated from Algebra i Logika, Vol. 48, No. 3, pp. 342-377, May-June, 2009. Original article submitted March 19,
2009.
0002-5232/09/4803-0193 c© 2009 Springer Science+Business Media, Inc. 193

AgroupG satisfies the big powers condition if every generic sequence in G is independent. Such groups
are called BP-groups. This form of the BP condition was brought in sight in [1, 2], where some elementary
properties of BP-groups are given. Note that BP-groups are torsion-free since any nontrivial element of G
forms a generic sequence. We can easily generalize the definition of a BP-group by considering only generic
sequences of elements of a particular type (“hyperbolic elements”) to accommodate torsion or to include
relatively hyperbolic groups. For the purposes of this paper, however, the narrow definition suffices, and
we so leave it like that.
A class of BP-groups is quite broad. For instance, it contains all torsion-free Abelian groups and
free groups. Furthermore, torsion-free hyperbolic groups, as well as all of their subgroups, share the BP
property [3]. Obviously, the BP property is local, that is, is inherited by subgroups (as distinct, for instance,
from hyperbolicity), which provides us with a method of constructing infinitely generated BP-groups. In
particular, direct limits of BP-groups are again BP-groups. Note that inverse limits of BP-groups also
have the BP property. To furnish some other examples of BP-groups, we recall that a group G separates
(discriminates) a group H if, for any nontrivial element h ∈ H (any finite subset of nontrivial elements
h
1
,...,h
k
∈ H), there exists a homomorphism φ : H → G such that h
φ

=1(h
φ
i

=1fori =1,...,k). It
is not hard to see that any group discriminated by a BP-group is itself a BP-group [4] (see also Prop. 2
below). In particular, fully residually free groups are BP-groups.
Previously, as mentioned, the big powers condition was used in various forms in many areas of group
theory. In [5, 6], for instance, the BP condition was employed (though implicitly) for solving one-variable
equations in free groups. In [7], big powers were applied to decide on solvability of equations in free groups
in the form of Bulitko’s lemma. In [8], a weaker form of the BP condition was defined explicitly and was
used to prove residual freeness of surface groups. In [3], a general version of the BP condition was defined
in terms of an equivalent separation condition (see Sec. 2), and it was proved that this condition holds
for torsion-free hyperbolic groups. In [1], it was stated that a Z[t]-completion F
Z[t]
of a free group F is a
BP-group. It turned out that a similar argument can be applied in studying Z[t]-completions of arbitrary
torsion-free hyperbolic groups (see [4]). The BP property was employed as a basic tool in [9], where it
was shown that every finitely generated fully residually free group acts freely on a Z
n
-tree. Lastly, the BP
argument was essentially used in [10] in dealing with the implicit function theorem for free groups.
In Sec. 2, we discuss equivalent definitions of the BP property and establish elementary properties of
BP-groups. In particular, it is shown that a BP-group satisfying a nontrivial identity is Abelian (Thm. 1).
This imposes some restrictions on subgroups of BP-groups.
In Sec. 3, we introduce the concept of being strongly isolated for subgroups, which is, in a sense, a
combinatorial counterpart of the geometric notion of being quasiconvex. Intuitively, a subgroup A is
strongly isolated in a group G if G satisfies BP modulo A (see Sec. 3). It turns out that centralizers of BP-
groups are strongly isolated, and so are quasiconvex isolated subgroups of torsion-free hyperbolic groups.
Also, using results of Sec. 5, we show that free factors of BP-groups are strongly isolated (see Prop. 7).
In Secs. 4-6, we prove the main result of the paper (Thm. 3), which states that under certain natural
conditions, the universal group U(P ) of Stallings’ pregroup P is a BP-group. In particular, free products
with amalgamation (some HNN-extensions) preserve the BP property provided that it holds in their factors
(base groups) and certain natural requirements on amalgamated (associated) subgroups are met. For
example, a free product of CSA BP-groups G
1
and G
2
with amalgamation along a subgroup C,whichis
strongly isolated and malnormal in G
1
and in G
2
, is a BP-group (Thm. 4). In particular, a free product of
CSA BP-groups is a BP-group (see Sec. 4).
194