Free groups and the group of their outer automorphisms have been extensively studied in analogy with (fundamental groups of) surfaces and the mapping class groups of surfaces. We study the analogue of intersection numbers of simple curves, namely the Scott-Swarup algebraic intersection number of splittings of a free group and we also study embedded spheres in $3$- manifold of the form $ M =\\sharp_n S^2 \\times S^1 $. The fundamental group of $M$ is a free group of rank $n$. This $3$-manifold will be our model for free groups. We construct geosphere laminations in free group which are analogues of geodesic laminations on a surface.

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Last updated: 27 Feb 2024