Generalized geodesic currents on free groups

Details Generalized geodesic currents on free groups Generalized geodesic currents on free groups

2011-05-28

arxiv.org/abs/1105.5742v1

Mathematics

Group Theory

Scientific paper

We introduce and study the space of \emph{generalized geodesic currents} on the free group $F_N$, which are measure-theoretic objects generalizing conjugacy classes of nontrivial finitely generated subgroups of $F_N$. The space $\mathcal GCurr(F_N)$ of generalized currents contains the space $Curr(F_N)$ of ordinary geodesic currents as a closed $Out(F_N)$-invariant subspace. Much of the theory of $Curr(F_N)$ naturally extends to the $\mathcal GCurr(F_N)$ context, but new dynamical, geometric and algebraic features also arise there. In particular, similarly to the case of ordinary currents, there is a continuous $Out(F_N)$-invariant "geometric intersection form" between the unprojectivized Outer space $cv_N$ and the space $\mathcal GCurr(F_N)$ of generalized currents. Given a tree $T\in cv_N$ and the "counting current" $\eta_H\in \mathcal GCurr(F_N)$ of a finitely generated nontrivial subgroup $H\le F_N$, the value $$ of the intersection form turns out to be equal to the co-volume of $H$, that is the volume of the metric graph $H\setminus T_H$, where $T_H\subseteq T$ is the unique minimal $H$-invariant subtree of $T$.

Affiliated with

Also associated with

No associations

Rating

Generalized geodesic currents on free groups does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Generalized geodesic currents on free groups, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Generalized geodesic currents on free groups will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-289474