Burnside Condition on Some Intersection Subgroups

arXiv:1604.08387v1 [math.AT] 28 Apr 2016

Hanieh Mirebrahimi∗, Fateme Ghanei
Department of Pure Mathematics, Center of Excellence in Analysis on Algebraic Structures,
Ferdowsi University of Mashhad,
P.O.Box 1159-91775, Mashhad, Iran.

Abstract
In this paper, using the notions graphs, core graphs, immersions and covering maps
of graphs, introduced by Stallings in 1983, we prove the Burnside condition for the
intersection of subgroups of free groups with Burnside condition.
Keywords: Graph, Fundamental group, Immersion and Covering theory, Burnside
condition.
2010 MSC: 05E15, 05E18, 55Q05, 57M10
1. Introduction and Motivation
In [3] J. Stallings studied on free groups by theory of graphs. He introduced the
concept of immersions of graphs, provided an algorithmic process to study on finitely
generated subgroups of free groups. Using these tools, he also gave an elegant proof
for Howson’s theorem ”if A and B are finitely generated subgroups of a free group,
then A ∩ B is finitely generated”. Moreover, using immersions of graphs and core
graphs (graphs with no trees hanging on) some mathematicians such as Everitt and
Gersten studied on H. Neumann’s inequality on the rank of A ∩ B (see [1] and [2]).
Stallings also in [3], introduced another notation called ”Burnside condition for a
subgroup”. In this paper, we focus on this notion, and using similar methods, we
prove that if A and B are finitely generated subgroups of a free group F , and A ∩ B
satisfies the Burnside condition in both A and B, then A ∩ B satisfies the Burnside
condition in A ∨ B, the subgroup of F generated by A ∪ B.
Corresponding author
Email addresses: h_mirebrahimi@um.ac.ir (Hanieh Mirebrahimi),
fatemeh.ghanei91@gmail.com (Fateme Ghanei)
∗

Preprint submitted to Topology and its Applications

April 12, 2018

2. Preliminaries
In this section, all our notations come from [3]. A graph X consists of two sets
E and V (edges and vertices), with three functions −1 : E −→ E and s, t : E −→ V
such that (e−1 )−1 = e, e−1 6= e, s(e−1 ) = t(e) and t(e−1 ) = s(e). We say that the
edge e ∈ E has initial vertex s(e) and terminal vertex t(e). The edge e−1 is also
called the reverse of e.
A map of graphs f : X −→ Y is a function which maps edges to edges and vertices
to vertices. Also we have f (e−1 ) = f (e)−1 , f (s(e)) = s(f (e)) and f (t(e)) = t(f (e)).
A path p in X of length n = |p|, with initial vertex u and terminal vertex v, is an
n-tuple of edges of X of the form p = e1 ...en such that for i = 1, ..., n − 1, we have
t(ei ) = s(ei+1 ) and s(e1 ) = u and t(en ) = v. For n = 0, given any vertex v, there
is a unique path Λv of length 0 whose initial and terminal vertices coincide and are
equal to v. A path p is called a circuit if its initial and terminal vertices coincide.
If p and q are paths in X and the terminal vertex of p equals the initial vertex
of q, they may be concatenated to form a path pq with |pq| = |p| + |q|, whose initial
vertex is that of p and whose terminal vertex is that of q.
A round-trip is a path of the form ee−1 . A reduced path is a path in X containing
no round-trip. An elementary reduction is insertion or deletion a round-trip in a
path. Two paths p and q are homotopic (written p ∼ q) iff there is a finite sequence
of elementary reductions taking one path to the other. Homotopic paths must have
the same start and terminal vertices and also, homotopy is an equivalence relation
on the set of paths with same start and same terminal vertices in X. Moreover, any
path in X is homotopic to a unique reduced path in X.
Let v be a fix vertex in X, π1 (X, v) is defined to be the set of all homotopy
classes of closed paths with initial and terminal vertex v. Then π1 (X, v) together
with the product [p][q] := [pq] forms a group with identity [Λv ] and inverse element
[p]−1 = [p−1 ].
For a fix vertex v in X, the star of v in X is defined as follows:
St(v, X) = {e ∈ E : s(e) = v}.
A map f : X −→ Y yields, for each vertex v ∈ X, a function fv : St(v, X) −→
St(f (v), Y ). If for each vertex v ∈ X, fv is injective, we call f an immersion and
if for each vertex v ∈ X, fv is surjective, we call f a locally surjective. If each fv is
bijective, we call f a covering map .
The theory of coverings of graphs is almost completely analogous to the topological theory of coverings. Immersions have some of the properties of coverings, one of
them we need it more, is the following one:
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”For a given finite set of elements {α1 , ..., αn } ⊆ π1 (X, u), there is a connected
graph Y and an immersion f : Y −→ X such that f∗ (π1 (Y )) = S, in which S is the
subgroup of π1 (X, u) generated by {α1 , ..., αn }”.
If G is a group, a G-graph X is a graph with an action of G on the left on X by
maps of graphs, such that for all g ∈ G and every edge e, ge 6= e−1 . In this case,
the quotient graph X/G, and the natural quotient map of graphs q : X → X/G are
defined. It is easy to see that, in general q : X → X/G is locally surjective.
We call G acts freely on X, whenever v is a vertex of X, g ∈ G, and gv = v, then
g = 1, the identity element of G. In this case, q : X → X/G is an immersion, and
hence is a covering map.
A translation of a map of graphs f : X → Y is a map g : X → X which is an
isomorphism of graphs and for which f g = f . The set of all translations of f forms
a group G(f ) which acts on X. If f is an immersion, and X is connected, then G(f )
acts freely on X.
For a connected graph X, the universal cover f : X̃ → X is a covering map with
X̃ is connected and π1 (X̃) is trivial.
Lemma 2.1. [3] If f : X̃ → X is a universal covering map, then G(f ) ∼
= π1 (X)
which acts freely, by covering translations, on X̃, and also f is isomorphic to the
quotient map q : X̃ → X̃/G, i.e., there exists an isomorphism ϕ : X̃/G → X such
that ϕq = f .
Theorem 2.2. [3] Let

g1

Z3 −−−→ Z1


 g2

f
1y
y
f2

Z2 −−−→ X.
be a pullback diagram of graphs, where f1 and f2 are immersions. Let v1 and v2 be
vertices in Z1 and Z2 that f1 (v1 ) = f2 (v2 ) = w. Let v3 be corresponding vertex in
Z3 . Define f3 = f1 g1 = f2 g2 : Z3 → X, and Si = fi∗ (π1 (Zi , vi )), for i = 1, 2, 3. Then
S3 = S1 ∩ S2 .
3. Main results
In this section, we deduce our main result. Before it, we recall some notes from
[3] which are essential in our proof. First, we note to the core graphs whose roles are
more important.
A cyclically reduced circuit in a graph X is a circuit p = e1 ...en , which is reduced
as a path and for which e1 6= e−1
n . A graph X is said to be a core-graph if X
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is connected, has at least one edge and each of its edges belongs to at least one
cyclically reduced circuit.
In a connected graph X with non-trivial fundamental group, an essential edge is
an edge belonging to some cyclically reduced circuit. The core of X consists of all
essential edges of X and all initial vertices of essential edges.
If X is a connected graph with non-trivial fundamental group, then the core X ′
of X is a core-graph. If v is a vertex of X ′ , then the homomorphism induced by
inclusion, π1 (X ′ , v) −→ π1 (X, v) is an isomorphism.
Another notion, we are dealing with, is the Burnside condition for subgroups. A
subgroup S of a group G satisfies the Burnside condition if for every g ∈ G, there
exists some positive integer n such that g n ∈ S.
Lemma 3.1. [3] (a) Let f : X −→ Y be a finite-sheeted covering of connected
graphs, and v a vertex of X. Then f∗ (π1 (X, v)) ⊆ π1 (Y, f (v)) satisfies the Burnside
condition.
(b) Let f : X −→ Y be an immersion of connected graphs, Y be a core-graph, v
a vertex of X, and f∗ (π1 (X, v)) ⊆ π1 (Y, f (v)) satisfy the Burnside condition. Then
f is a covering map.
Finally, using all the above notes, we establish our main result in the follow.
Theorem 3.2. Let S1 and S2 be finitely generated subgroups of a free group F . If
S1 ∩ S2 satisfies the Burnside condition both in S1 and S2 , then S1 ∩ S2 also satisfies
the Burnside condition in S1 ∨ S2 , the subgroup generated by S1 ∪ S2 .
Proof. Similar to the argument of 7.8 in [3], we start with the following pullback
diagram of immersions
g1
Z3 −−−→ Z1


 g2

f
1
y
y
f2

Z2 −−−→ X.
where f1 and f2 are immersions, Z3 is a subgraph of the pullback; Z1 , Z2 and Z3 are
core-graphs; f3 = f1 g1 = f2 g2 ; v3 is a vertex of Z3 , v1 and v2 are the images of v3 in
Z1 and Z2 ; w = fi (vi ) (i = 1, 2, 3), F = π1 (X, w), Si = fi∗ (π1 (Zi , vi )) (i = 1, 2, 3).
Using Theorem 2.2, S3 = S1 ∩ S2 .
By Lemma 3.1 (b), since S3 satisfies the Burnside condition both in S1 and S2 ,
it follows that the immersions g1 and g2 are coverings.
Let r : Z̃3 → Z3 be a universal covering, and consider g̃1 = g1 r and g̃2 = g2 r, which
are consequently universal coveings. Then by Lemma 2.1, G(g̃1 ) = S1 , G(g̃2 ) = S2
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and the universal coverings g̃1 : Z̃3 → Z1 and g̃2 : Z̃3 → Z2 are isomorphic to
quotient maps q1 : Z̃3 → Z̃3 /G(g̃1) and q2 : Z̃3 → Z̃3 /G(g̃2 ), respectively, i.e., there
are isomorphisms ϕ1 : Z̃3 /G(g̃1 ) → Z1 and ϕ2 : Z̃3 /G(g̃2) → Z2 such that ϕ1 q1 = g̃1
and ϕ2 q2 = g̃2 . By definition, for any covering transformation σ ∈ G(g̃1 ) = S1 ,
g̃1 σ = g̃1 , and so g1 rσ = g1 r. Hence, f3 rσ = f3 r = h. So σ is a transformation of
the immersion h. Similarly, any covering transformation of g̃2 is a transformation of
h. Now, suppose K is the group of transformations of h generated by G(g̃1 ) ∪ G(g̃2 ),
then K ∼
= S1 ∨ S2 and we have the following pushout diagram.
q1

/ Z̃3 /G(g̃1 )
▼▼▼
▼▼▼t3
q2
t1
▼▼▼
▼&


/ Z̃3 /K
Z̃3 /G(g̃2 )

Z̃3 ▼▼

t2

Therefore, because of the universal property of pushout, for graph X and maps
f1 ϕ1 and f2 ϕ2 there exists a unique map s : Z̃3 /K → X such that st1 = f1 ϕ1 and
st2 = f2 ϕ2 .
Since h is an immersion, then G(h) and hence K acts freely on Z̃3 , and so t3 is a
covering. It follows that t1 is a covering too. Z̃3 /G(g̃1 ) is a finite graph and so t1 is a
finite-sheeted covering. Thus by Lemma 3.1 (a), t1∗ (π1 (Z̃3 /G(g̃1 ))) satisfies the Burnside condition in π1 (Z̃3 /K); so s∗ (t1∗ (π1 (Z̃3 /G(g̃1)))) = f1 ∗ (ϕ1 ∗ (π1 (Z̃3 /G(g̃1 )))) =
f1 ∗ (π1 (Z1 )) = S1 satisfies the Burnside condition in s∗ (π1 (Z̃3 /K)). As we can see,
s∗ (π1 (Z̃3 /K)) contains S1 and similarly S2 and hence it contains S1 ∨ S2 . Therefore, since S1 ⊆ S1 ∨ S2 ⊆ s∗ (π1 (Z̃3 /K)) and S1 satisfies the Burnside condition in
s∗ (π1 (Z̃3 /K)), then S1 also satisfies the Burnside condition in S1 ∨ S2 . Now, by the
fact that S1 ∩ S2 satisfies the Burnside condition in S1 , the result holds.
References
References
[1] B. Everitt, Graphs, free groups and the Hanna Neumann conjecture, Journal of Group theory, 11 (2008), 885-899.
[2] S. M. Gersten, Intersections of finitely generated subgroups of free groups
and resolutions of graphs, Invent. Math. 71 (1983), 567–591.
[3] J. R. Stallings, Topology of finite graphs, Invent. math. 71 (1983), 551-565.

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