On the embedding problem for generalized Baumslag-Solitar
groups
F. Dudkin
A finitely generated group \(G\) is called a generalized
Baumslag–Solitar group (GBS group) if \(G\) is acting on a tree
\(T\) with infinite cyclic edge and vertex stabilizers. Then, by
the Bass–Serre theorem, \(G\) is a \(\pi_1(\mathbb{A})\)–
fundamental group of some graph of groups \(\mathbb{A}\) (see, for
example, [1]), with vertex and edge groups infinite cyclic.
A GBS group can be described by a labeled graph
\(\mathbb{A}=(\Gamma, \lambda)\), where \(\Gamma\) is a finite graph
and \(\lambda\colon E(\Gamma)\to\mathbb{Z}\setminus\{0\}\) is a
labeling of edges of \(\Gamma\). The label \(\lambda(e)\) written on an edge
\(e\) with origin \(v\) determines the embedding \(\alpha_e\colon e\to
v^{\lambda(e)}\) of cyclic edge group \(\langle e\rangle\) into
cyclic vertex group \(\langle v\rangle\).
GBS groups have been studied from different positions [2],
[3], [4]. In particular, the isomorphism problem for GBS groups
was discussed: to define algorithmically when two given labeled
graphs set isomorphic GBS groups. In spite of the fact that in
some particular cases the isomorphism problem was solved [5], [6],
[7], in general, the existence of an algorithm was not established.
We study the embedding problem for GBS groups: to define
algorithmically when two labeled graphs
\(\mathbb{A}_1=(\Gamma_1,\lambda_1)\) and
\(\mathbb{A}_2=(\Gamma_2,\lambda_2)\) define GBS groups such that
\(\pi_1(\mathbb{A}_1)\) is embeddable into \(\pi_1(\mathbb{A}_2)\).
There are two basic deformations of labeled graph that do not
change it's fundamental groups – expansions and collapses:
A labeled graph \(\mathbb{A}\) is reduced if it does not admit a
collapse move. An elementary deformation is a finite sequence of
collapse and expansion moves. Given a labeled graph \(\mathbb{A}\),
the deformation space \(\mathcal{D}_\mathbb{A}\) of \(\mathbb{A}\) is
the set of all labeled graphs related to \(\mathbb{A}\) by an
elementary deformation. We prove
Theorem.Let \(\mathbb{A}_1, \mathbb{A}_2\) be labeled
graphs. If there is only finitely many reduced labeled graphs in
\(\mathcal{D}_{\mathbb{A}_1}\) then the embedding problem
\(\pi_1(\mathbb{A}_1)\to \pi_1(\mathbb{A}_2)\) is decidable.
References
J. P. Serre, Trees, Springer-Verlag, Berlin-New York (1980), 142 p.
M. Clay, M. Forester, Whitehead moves for \(G\)-trees, Bull. London Math. Soc., 41, N2 (2009),
205–212.
M. Forester, On uniqueness of JSJ decomposition of finitely generated groups, Comm. Math. Helv., 78 (2003), 740–751.
M. Clay, Deformation spaces of \(G\)–trees and automorphisms of Baumslag–Solitar
groups, Groups Geom. Dyn., 3 (2009), 39–69.
M. Forester, Splittings of generalized Baumslag-Solitar groups, Geom. Dedicata, 121, N1 (2006),
43–59.
M. Clay, M. Forester, On the isomorphism problem for generalized Baumslag-Solitar groups, Algebr. Geom. Topol., 8 N4 (2008), 2289–2322.
G. Levitt, On the automorphism group of generalized Baumslag-Solitar groups, Geom. Topol.,
11 (2007), 473–515.
