Length functions on groups and rigidity

Abstract

Let $G$ be a group. A function $l:G\rightarrow \lbrack 0,\infty )$ is called a length function if (1) $l(g^{n})=|n|l(g)$ for any $g\in G$ and $n\in \mathbb{Z};$ (2) $l(hgh^{-1})=l(g)$ for any $h,g\in G;$ and (3) $l(ab)\leq l(a)+l(b)$ for commuting elements $a,b.$ Such length functions exist in many branches of mathematics, mainly as stable word lengths, stable norms, smooth measure-theoretic entropy, translation lengths on $\mathrm{CAT}(0)$ spaces and Gromov $\delta $% -hyperbolic spaces, stable norms of quasi-cocycles, rotation numbers of circle homeomorphisms, dynamical degrees of birational maps and so on. We study length functions on Lie groups, Gromov hyperbolic groups, arithmetic subgroups, matrix groups over rings and Cremona groups. As applications, we prove that every group homomorphism from an arithmetic subgroup of a simple algebraic $\mathbb{Q}$-group of $\mathbb{Q}$-rank at least $2,$ or a finite-index subgroup of the elementary group $E_{n}(R)$ $(n\geq 3)$ over an associative ring, or the Cremona group $\mathrm{Bir}(P_{\mathbb{C}}^{2})$ to any group $G$ having a purely positive length function must have its image finite. Here $G$ can be outer automorphism group $\mathrm{Out}(F_{n})$ of free groups, mapping classes group $\mathrm{MCG}(\Sigma_{g})$, $\mathrm{CAT}% (0)$ groups or Gromov hyperbolic groups, or the group $\mathrm{Diff}(\Sigma ,\omega )$ of diffeomorphisms of a hyperbolic closed surface preserving an area form $\omega .$