Commensurability of Fuchsian Groups and Their Axes

Theorem. For each arithmetic Fuchsian group Γ, there exists an infinite order elliptic element e such that e(ax(Γ)) = ax(Γ). Recall that a Fuchsian group is a discrete subgroup of PSL2(R) ∼= isom(H). We denote by ax(Γ) the set of axes of hyperbolic elements of the Fuchsian group Γ. The proof follows easily from known properties of arithmetic Fuchsian groups. Recall that an arithmetic Fuchsian group is constructed as follows (see [5]). Let k be a totally real number field, and A a quaternion algebra over k which is ramified at all places of k except one, which we may take to be the identity. Let ρ be the embedding of A into M2(R) induced by splitting A at the identity place of k. Let O be an order of A, and O the elements of reduced norm one in O. Then ρ(O) is a discrete subgroup of SL2(R) and its projection to PSL2(R) via P is a finitely generated Fuchsian group of the first kind. A Fuchsian group Γ is arithmetic if it is commensurable with some such group Pρ(O). Subgroups Γ1 and Γ2 of PSL2(R) are commensurable if Γ1∩Γ2 has finite index in both Γ1 and Γ2. Define the commensurator comm(Γ) of a subgroup Γ of PSL2(R) by comm(Γ) = {φ ∈ PSL2(R) | φΓφ−1 is commensurable to Γ}. It is a theorem of Margulis [3] that comm(Γ) is non-discrete if and only if Γ is arithmetic. In particular, since it is non-elementary and non-discrete, comm(Γ) must contain an infinite order elliptic e [2]. It is easy to verify that, if Γ1 and Γ2 are commensurable, then ax(Γ1) = ax(Γ2). Since e ∈ comm(Γ), eΓe−1 and Γ are commensurable, and so ax(Γ) = ax(eΓe−1) = e(ax(Γ)). In fact, the Theorem holds true for any arithmetic subgroup Γ of isom(H). Briefly, for such groups, Margulis’ result still holds, and Abikoff and Haas [1] have shown that a non-elementary subgroup of isom(H) which does not keep a proper subspace invariant is discrete if and only if it contains no infinite order elliptic elements. Since Γ arithmetic implies H/Γ has finite volume, Γ cannot keep