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We show that every continuous homogeneous quasimorphism on a finite-dimensional 1-connected simple Lie group arises as the relative growth of any continuous bi-invariant partial order on that group. More generally we show, that an arbitrary homogeneous quasimorphism can be reconstructed as the relative growth of a partial order subject to a certain sandwich condition. This provides a link between invariant orders and bounded cohomology and allows the concrete computation of relative growth for finite dimensional simple Lie groups as well as certain infinite-dimensional Lie groups arising from symplectic geometry.
We prove the vanishing of the cup product of the bounded cohomology classes associated to any two Brooks quasimorphisms on the free group. This is a consequence of the vanishing of the square of a universal class for tree automorphism groups.
We study the construction of quasimorphisms on groups acting on trees introduced by Monod and Shalom, that we call median quasimorphisms, and in particular we fully characterise actions on trees that give rise to non-trivial median quasimorphisms. Ro
We define toric partial orders, corresponding to regions of graphic toric hyperplane arrangements, just as ordinary partial orders correspond to regions of graphic hyperplane arrangements. Combinatorially, toric posets correspond to finite posets und
Let $F_n$ be a free group of finite rank $n geq 2$. We prove that if $H$ is a subgroup of $F_n$ with $textrm{rk}(H)=2$ and $R$ is a retract of $F_n$, then $H cap R$ is a retract of $H$. However, for every $m geq 3$ and every $1 leq k leq n-1$, there
The deck of a graph $G$ is the multiset of cards ${G-v:vin V(G)}$. Myrvold (1992) showed that the degree sequence of a graph on $ngeq7$ vertices can be reconstructed from any deck missing one card. We prove that the degree sequence of a graph with av