Bridgeman's orthospectrum identity

We give a short derivation of an identity of Bridgeman concerning orthospectra of hyperbolic surfaces.Comment: 5 pages, 3 figures; v3 minor errors correcte

Scl, sails and surgery

We establish a close connection between stable commutator length in free groups and the geometry of sails (roughly, the boundary of the convex hull of the set of integer lattice points) in integral polyhedral cones. This connection allows us to show that the scl norm is piecewise rational linear in free products of Abelian groups, and that it can be computed via integer programming. Furthermore, we show that the scl spectrum of nonabelian free groups contains elements congruent to every rational number modulo $\mathbb{Z}$, and contains well-ordered sequences of values with ordinal type $\omega^\omega$. Finally, we study families of elements $w(p)$ in free groups obtained by surgery on a fixed element $w$ in a free product of Abelian groups of higher rank, and show that \scl(w(p)) \to \scl(w) as $p \to \infty$.Comment: 23 pages, 4 figures; version 3 corrects minor typo

Certifying incompressibility of non-injective surfaces with scl

Cooper-Manning and Louder gave examples of maps of surface groups to PSL(2,C) which are not injective, but are incompressible (i.e. no simple loop is in the kernel). We construct more examples with very simple certificates for their incompressibility arising from the theory of stable commutator length.Comment: 5 pages; version 2 incorporates referee's suggestion

Dynamical forcing of circular groups

In this paper we introduce and study the notion of dynamical forcing. Basically, we develop a toolkit of techniques to produce finitely presented groups which can only act on the circle with certain prescribed dynamical properties. As an application, we show that the set X ⊂ R/Z consisting of rotation numbers θ which can be forced by finitely presented groups is an infinitely generated Q-module, containing countably infinitely many algebraically independent transcendental numbers. Here a rotation number θ is forced by a pair (G_θ, α), where G_θ is a finitely presented group G_θ and α ∈ G_θ is some element, if the set of rotation numbers of ρ(α) as ρ ∈ Hom(G_θ, Homeo^(+)(S^1)) is precisely the set {0,±θ}. We show that the set of subsets of R/Z which are of the form rot(X(G, α)) = {r ∈ R/Z | r = rot(ρ(α)), ρ ∈ Hom(G, Homeo^(+)(S^1))}, where G varies over countable groups, are exactly the set of closed subsets which contain 0 and are invariant under x→−x. Moreover, we show that every such subset can be approximated from above by rot(X(G_i, α_i)) for finitely presented G_i. As another application, we construct a finitely generated group Γ which acts faithfully on the circle, but which does not admit any faithful C^1 action, thus answering in the negative a question of John Franks