Tag - Group theory
We prove that generic elements of braid groups are pseudo-Anosov, in the following sense: in the Cayley graph of the braid group with n≥3 strands, with respect to Garside's generating set, we prove that the proportion of pseudo-Anosov braids in the ball of radius l tends to 1 exponentially quickly as l tends to infinity. Moreover, with a similar notion of genericity, we prove that for generic pairs of elements of the braid group, the conjugacy search problem can be solved in quadratic time. The idea behind both results is that generic braids can be conjugated ''easily'' into a rigid braid.
For a geometrically finite hyperbolic group with small critical exponent, the spectral
method for counting is not available, as there is no point eigenvalue of the Laplace operator on the L2-spectrum. We will explain counting results for orbits of a big class of thin groups acting on a symmetric variety of the real hyperbolic group, which are obtained via ergodic approach.
We will describe a recent effective counting result for Apollonian circle packings. The main ingredient of this result is an effective equidistribution of closed horospheres in an infinite volume hyperbolic 3-manifold whose fundamental group has critical exponent bigger than one. We will explain how the spectral theory of Lax and Phillips can be used for such equidistribution results.
A two-hour course on expanders, thin subgroups of Lie groups, and superstrong approximation.
An elementary homomorphism from a free group to the pure braid group yields interesting connections between braid groups, homotopy theory, and low dimensional topology. This map induces a map on the Lie algebra obtained from the descending central series. Further, this map induces a morphism of simplicial groups. All of these maps are shown to be injective.
Brunnian braids are discussed. The analogous maps of Lie algebras induced on the filtration quotients of the mod-p descending central series is again an injection. Using these facts it turns out that the homotopy groups of this simplicial group, those of the 2-sphere, are isomorphic to natural subquotients of the pure braid group. In addition, the mod-p analogues give a connection between the classical unstable Adams spectral sequence, and the mod-p analogues of Vassiliev invariants of pure braids.
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