Loxodromic elements in big mapping class groups via the Hooper-Thurston-Veech construction

Abstract

Let S be an infinite-type surface and p ∈ S . We show that the Thurston-Veech construction for pseudo-Anosov elements, adapted for infinite-type surfaces, produces infinitely many loxodromic elements for the action of Mod(S ; p) on the loop graph L(S ; p) that do not leave any finite-type subsurface S ′ ⊂ S invariant. Moreover, in the language of [BW18b], Thurston-Veech's construction produces loxodromic elements of any weight. As a consequence of Bavard and Walker's work, any subgroup of Mod(S ; p) containing two "Thurston-Veech loxodromics" of different weight has an infinite-dimensional space of non-trivial quasimorphisms.

An infinite clique of high-filling rays in the plane minus a Cantor set

“…But because this construction uses only finite type subsurfaces, these cliques are not specific to infinite type surfaces. However, in [6], Israel Morales and Ferrán Valdez constructed loxodromic elements which do not preserve any finite type subsurface and which have any chosen finite number of high-filling rays in their cliques. These high-filling rays are thus specific to infinite type surfaces.…”