On Fuchsian Groups with the Same Set of Axes

Abstract

Let Γ be a Fuchsian group. We denote by ax(Γ) the set of axes of hyperbolic elements of Γ. Define Fuchsian groups Γ1 and Γ2 to be isoaxial if ax(Γ1) = ax(Γ2). The main result in this note is to show (see Section 2 for definitions) the following. THEOREM 1.1. Let Γ1 and Γ2 be isoaxial arithmetic Fuchsian groups. Then Γ1 and Γ2 are commensurable. This result was motivated by the results in [6], where it is shown that if Γ1 and Γ2 are finitely generated non‐elementary Fuchsian groups having the same non‐empty set of simple axes, then Γ1 and Γ2 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(Γ) and the subgroup of PGL(2, R) which preserves the set ax(Γ) by Σ(Γ) (careful definitions are given below), we have the following theorem which describes the commensurability subgroup of an arithmetic Fuchsian group geometrically. THEOREM 2.4. Let Γ be an arithmetic Fuchsian group. Then Comm(Γ) = Σ(Γ). The problem of ‘isoaxial implies commensurable’ is like a dual problem to ‘isospectral implies commensurable’. It was shown in [8] that this latter statement holds for arithmetic Fuchsian groups. As in the case considered here, the general statement about isospectrality implying commensurability is still open. 1991 Mathematics Subject Classification 20H10.

Geodesic intersections and isoxial Fuchsian groups.

“…In the case of Fuchsian groups of the first kind Long and Reid [4] proved the conjecture for arithemetic groups. Also notice that if Γ 1 and Γ 2 are isoaxial Fuchsian groups, then for any γ ∈ Γ 2 ax(Γ 1 ) = ax(γΓ 1 γ −1 ), and therefore γ ∈ Aut(ax(Γ)).…”

“…In the case of Fuchsian groups of the first kind Long and Reid [4] proved the conjecture for arithemetic groups.…”

Pseudomodular surfaces

“…Recall that two Fuchsian groups are defined to be isoaxial if they share the same set of axes in H 2 . If both the Fuchsian groups are assumed arithmetic, then [12] shows they are commensurable (without conjugating). There is considerable experimental evidence that the pseudomodular groups we know of are not isoaxial with SLð2; ZÞ; for example it would appear that if A denotes the axis of the element but M k is in Dð5=7; 6Þ, then k e 36 and 4jk.…”

Limit Sets and Commensurability of Kleinian Groups

“…In 1990, G. Mess [11] showed that if G 1 and G 2 are non-elementary finitely generated Fuchsian groups having the same nonempty set of simple axes, then G 1 and G 2 are commensurable. Using some technical results on arithmetic Kleinian groups, D. Long and A. Reid [9] gave an affirmative answer to this question in the case where G 1 and G 2 are arithmetic Kleinian groups. Note that all the confirmed cases for the question are geometrically finite groups.…”