Date of Submission


Date of Award


Institute Name (Publisher)

Indian Statistical Institute

Document Type

Doctoral Thesis

Degree Name

Doctor of Philosophy

Subject Name



Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)


Gadgil, Siddhartha

Abstract (Summary of the Work)

Topological and geometric methods have played a major role in the study of infinite groups since the time of Poincar´e and Klein, with the work of Nielsen, Dehn, Stallings and Gromov showing particularly deep connections with the topology of surfaces and three-manifolds. This is in part because a surface or a 3-manifold is essentially determined by its fundamental group, and has a geometric structure due to the Poincar´e-K¨obe-Klein uniformisation theorem for surfaces and Thurston’s geometrisation conjecture, which is now a theorem of Perelman, for 3-manifolds.A particularly fruitful instance of such an interplay is the relation between intersection numbers of simple curves on a surface and the hyperbolic geometry and topology of the surface. This has reached its climax in the classification of finitely generated Kleinian groups by Yair Minsky and his collaborators, who along the way developed a deep understanding of the geometry of the curve complex.Free (nonabelian) 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.In my thesis, 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 = ]nS2 × S1 . 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.Chapter 1 In this chapter, we introduce basic concepts related to free product, free groups and splittings of groups.Chapter 2 In this chapter, we study geometric intersection number of simple closed curves on a surface. In particular, we see its applications to study geometric properties of curve complex of the surface. We also study topological properties of curve complex. We shall see how curve complex is used to study mapping class group of surfaces. The geometric intersection number has been used to study Thurston’s compactification of Teichm¨uller space of surface and the boundary of Teichm¨uller space, namely the space of projectivized measured laminations. At the end of this chapter, we study its analogue sphere complex of a 3-manifold and its topological properties.Chapter 3 In this chapter, we study the model 3-manifold M = ]kS 2 × S 1 . We also see how a partition of ends of the space Mf, the universal cover of M, corresponds to an embedded spheres in Mf. We also discuss the intersection number of a proper path in Mf with a homology class in H2(Mf). At the end of this chapter, we see how embedded spheres in M correspond to splittings of the fundamental group of M.Chapter 4 Scott and Swarup [39] introduced an algebraic analogue, called the algebraic intersection number, for a pair of splittings of groups. This is based on the associated partition of the ends of a group [42].


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