Infinitely many lattice surfaces with special pseudo-Anosov maps

Abstract

We give explicit pseudo-Anosov homeomorphisms with vanishing Sah-Arnoux-Fathi invariant. Any translation surface whose Veech group is commensurable to any of a large class of triangle groups is shown to have an affine pseudo-Anosov homeomorphism of this type. We also apply a reduction to finite triangle groups and thereby show the existence of non-parabolic elements in the periodic field of certain translation surfaces.

Some Normal Subgroups of Extended Generalized Hecke Groups

“…Also, all Hecke groups H q are included in generalized Hecke groups H p;q . Also, generalized Hecke groups H p;q have been studied extensively for many aspects in the literature (for examples, please see, [3], [4], [5], [6], [7] and [8]).…”

Commutator Subgroups of Generalized Hecke and Extended Generalized Hecke Groups

“…Furthermore all Hecke groups H q are included in generalized Hecke groups H p,q . Generalized Hecke groups H p,q have been also studied by Calta and Schmidt in [2] and [3].…”

“…. Also, the signatures of H 2,q is either ( q−1 2 ; ∞) if q is odd, or ( q−2 2 ; ∞ (2) ) if q is even. These results coincide with the ones given in [1] and [20], for Hecke groups H q .…”

On Normal Subgroups of Generalized Hecke Groups

“…Calta and Schmidt defined a continued fraction algorithm for groups H 3,q to show various properties of this group [15]. They studied Veech groups commensureble with generalized Hecke groups and pseudo-Anasov maps in [16].…”

“…In [16] where a, b, c, d are elements of the trace field K p,q = Q(λ p /2, λ q /2). These elements are named even and odd respectively.…”