Isoscattering on Surfaces

In this paper, we give a number of examples of pairs of non-compact surfaces S 1 and S 2 which are isoscattering, to be defined below. Our basic construction is based on a version of Sunada's Theorem [Su], which has been refined using the technique of transplantation ([Be], [Zel]) so as to be applicable to isoscattering. See [BGP] and [BP] for this approach, which is reviewed below. Our aim here is to present a number of examples which are exceptionally simple in one or more senses. Thus, the present paper can be seen as an extension of [BP], where the aim was to construct isoscattering surfaces with precisely one end. We will show: Theorem 0.1 (a) There exist surfaces S 1 and S 2 of genus 0 with eight ends which are isoscattering. (b) There exist surfaces S 1 and S 2 of constant curvature −1 which are of genus 0 and have fifteen ends. (c) There exist surfaces S 1 and S 2 of genus 1 with five ends, or genus 2 with three ends, which are isoscattering. (d) There exist surfaces S 1 and S 2 of constant curvature −1 which are of genus 1 with thirteen ends, or genus 2 with five ends, or genus 3 with three ends, which are isoscattering. (e) There exist surfaces S 1 and S 2 of genus 3 with one end, or constant curvature of genus 4 with one end, which are isoscattering. Part (e) is just a statement of the results of [BP], and is recorded here for the sake of completeness. It will not be discussed further in this paper. The nature of the ends in Theorem 0.1 is not too important. In the cases where the curvature is variable, they can be taken to be hyperbolic funnels or Euclidean cones, or to be hyperbolic finite-area cusps. In the constant curvature −1 cases, they can be taken either to be infinite-area funnels or finite-area cusps. Recall that a surface S is called a congruence surface if S = H 2 /Γ, where Γ is contained in P SL(2, Z) and contains a subgroup Γ k = a b c d ≡ ± 1 0 0 1 (mod k) 1 for some k. In other words, the group Γ is the inverse image of a subgroup of P SL(2, Z/k) under the natural map Theorem 0.2 There exist two congruence …