Summary: ON FUCHSIAN GROUPS WITH THE SAME SET OF AXES
D. D. LONG and A. W. REID
Let be a Fuchsian group. We denote by ax() the set of axes of hyperbolic
elements of . Define Fuchsian groups 1 and 2 to be isoaxial if ax(1) = ax(2).
The main result in this note is to show (see Section 2 for definitions) the following.
Theorem 1.1. Let 1 and 2 be isoaxial arithmetic Fuchsian groups. Then 1
and 2 are commensurable.
This result was motivated by the results in , where it is shown that if 1
and 2 are finitely generated non-elementary Fuchsian groups having the same non-
empty set of simple axes, then 1 and 2 are commensurable. The general question
of isoaxial was left open. Theorem 1.1 is therefore a partial answer. Arithmetic
Fuchsian groups are a very special subclass of Fuchsian groups (for example, there
are at most finitely many conjugacy classes of such groups of a fixed signature), and
the general case at present seems much harder to resolve.
The technical result of this paper which implies Theorem 1.1 is Theorem 2.4
(below), proved in Section 4. Denoting the commensurability subgroup by Comm()
and the subgroup of PGL(2, R) which preserves the set ax() by () (careful
definitions are given below), we have the following theorem which describes the
commensurability subgroup of an arithmetic Fuchsian group geometrically.