Quasi-homomorphisms

Quasi-homomorphisms on Mapping Class Groups

We refine the construction of quasi-homomorphisms on mapping class groups. It is useful to know that there are unbounded quasihomomorphisms which are bounded when restricted to particular subgroups since then one deduces that the mapping class group is not boundedly generated by these subgroups. In this note we enlarge the class of such subgroups. The generalization is motivated by consideratio...

Failure of Separation by Quasi-homomorphisms in Mapping Class Groups

We show that mapping class groups of surfaces of genus at least two contain elements of infinite order that are not conjugate to their inverses, but whose powers have bounded torsion lengths. In particular every homogeneous quasi-homomorphism vanishes on such an element, showing that elements of infinite order not conjugate to their inverses cannot be separated by quasi-homomorphisms. This note...

Well quasi-order in combinatorics: embeddings and homomorphisms

Minimum Cost Homomorphisms to Locally Semicomplete and Quasi-Transitive Digraphs

For digraphs G and H , a homomorphism of G to H is a mapping f : V (G)→V (H) such that uv ∈ A(G) implies f(u)f(v) ∈ A(H). If, moreover, each vertex u ∈ V (G) is associated with costs ci(u), i ∈ V (H), then the cost of a homomorphism f is ∑ u∈V (G) cf(u)(u). For each fixed digraph H , the minimum cost homomorphism problem for H , denoted MinHOM(H), can be formulated as follows: Given an input di...

Quasi-homomorphisms and Stable Lengths in Mapping Class Groups

We give elementary applications of quasi-homomorphisms to growth problems in groups. A particular case concerns the number of torsion elements required to factorise a given element in the mapping class group of a surface.