Vietoris-Rips Complexes and Geometric Group Theory

Explore the role of Vietoris-Rips complexes in geometric group theory and their impact on studying topological finiteness properties of groups in this 51-minute talk. Delve into Rips' famous proof of eventual contractibility of Vietoris-Rips complexes in hyperbolic groups, demonstrating their geometric actions on contractible complexes. Examine Alonso's use of these complexes to establish quasi-isometry invariance of finiteness properties type FP_n and F_n. Discover recent Morse theoretic techniques for proving contractibility in more general groups and spaces, including implications for asymptotically CAT(0) groups. Investigate the connection between type FP_n finiteness properties and the (non-)persistent homology of Vietoris-Rips complexes, highlighting an area ripe for further exploration in group theory.