Title of Seminar: Infosys Chandrasekharan Random Geometry Colloquium
Title of Talk: Existence of train tracks for free group automorphisms
Speaker: Indranil Bhattacharyya, TIFR, Mumbai
Date: November 11, 2024
Time: 15:15:00 Hours
Venue: A-369
Abstract: Train tracks were initially introduced by Thurston in the classification of surface automorphisms. Later, the idea was generalized in case of free groups by Bestvina and Handel to keep track of the direction of the edges in a graph. In this talk, we will prove the following theorem of Bestvina -? Every irreducible automorphism of the free group can be realized topologically by a train track map?. Bers classified surface automorphisms using Teichmuller metric. In the same analogy, we will introduce a metric space called Outer space and define an asymmetric metric on it. Then we shall classify elliptic, parabolic and hyperbolic automorphisms to prove the theorem. The existence of a train track map helps us to understand the growth of the conjugacy class of a certain group element under a free group automorphism.
Title of Seminar: Algebraic Geometry Preprint Seminar
Title of Talk: Free summands of stably free modules.
Speaker: Sandeep S, TIFR, Mumbai
Date: November 11, 2024
Time: 16:00:00 Hours
Venue: AG-77
Abstract: In this preprint by Ben Williams and W S Gant, https://arxiv.org/abs/2409.15445, the authors show that a stably free module $P$ over a ring $R$ satisfying $P \oplus R \cong R^{24m}$ for an integer $m$ has a free direct summand of rank 2 when $\mathbb{Q}\subset R$. This gives a partial converse to a theorem of Raynaud. The proof uses the methods of $\mathbb{A}^1$- homotopy theory. We will state the main results and outline the proof.
Title of Talk: Existence of train tracks for free group automorphisms
Speaker: Indranil Bhattacharyya, TIFR, Mumbai
Date: November 11, 2024
Time: 15:15:00 Hours
Venue: A-369
Abstract: Train tracks were initially introduced by Thurston in the classification of surface automorphisms. Later, the idea was generalized in case of free groups by Bestvina and Handel to keep track of the direction of the edges in a graph. In this talk, we will prove the following theorem of Bestvina -? Every irreducible automorphism of the free group can be realized topologically by a train track map?. Bers classified surface automorphisms using Teichmuller metric. In the same analogy, we will introduce a metric space called Outer space and define an asymmetric metric on it. Then we shall classify elliptic, parabolic and hyperbolic automorphisms to prove the theorem. The existence of a train track map helps us to understand the growth of the conjugacy class of a certain group element under a free group automorphism.
Title of Seminar: Algebraic Geometry Preprint Seminar
Title of Talk: Free summands of stably free modules.
Speaker: Sandeep S, TIFR, Mumbai
Date: November 11, 2024
Time: 16:00:00 Hours
Venue: AG-77
Abstract: In this preprint by Ben Williams and W S Gant, https://arxiv.org/abs/2409.15445, the authors show that a stably free module $P$ over a ring $R$ satisfying $P \oplus R \cong R^{24m}$ for an integer $m$ has a free direct summand of rank 2 when $\mathbb{Q}\subset R$. This gives a partial converse to a theorem of Raynaud. The proof uses the methods of $\mathbb{A}^1$- homotopy theory. We will state the main results and outline the proof.