Finiteness properties, subgroups of hyperbolic groups, and complex hyperbolic lattices

Abstract: Hyperbolic groups form an important class of finitely generated groups that has attracted much attention in geometric group theory. We call a group of finiteness type \(F_n\) if it has a classifying space with finitely many cells of dimension at most \(n\). This generalises finite presentability, which is equivalent to type \(F_2\). Hyperbolic groups are of type \(F_n\) for all \(n\). It is natural to ask if subgroups of hyperbolic groups inherit these strong finiteness properties. We use methods from complex geometry to show that every uniform arithmetic lattice with positive first Betti number in \(\mathrm{PU}(n, 1)\) admits a finite index subgroup, which maps onto the integers with kernel of type \(F_{n−1}\) but not \(F_n\). This answers an old question of Brady and produces many finitely presented non-hyperbolic subgroups of hyperbolic groups. This is joint work with Pierre Py.

algebraic topologydifferential geometrydynamical systemsgroup theorygeometric topologysymplectic geometry

Audience: researchers in the topic

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Geometry and topology online

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The talks start at 13:30. Talks are typically fifty minutes long, with ten minutes for questions.

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