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Hyperbolic triangular buildings with and without periodic planes

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Riikka Kangaslampi
Aalto Univ., Espoo, Finland/MPIM
Thu, 30/07/2015 - 16:30 - 17:30
MPIM Lecture Hall

In this talk we consider surface subgroups of groups acting simply
transitively on vertex sets of certain hyperbolic triangular
buildings. The study is motivated by Gromov's famous surface subgroup
question: Does every one-ended hyperbolic group contain a subgroup
which is isomorphic to the fundamental group of a closed surface of
genus at least 2? The question remains unanswered, even though copious
partial results exist.

First we will briefly explain the idea of the construction of groups
acting simply transitively on the vertices of hyperbolic triangular
buildings with the minimal generalized quadrangle as the link. Then
we'll consider surface subgroups of the obtained 23 torsion free
groups. We will show, that in most of the groups there are no periodic
apartments invariant under an action of a genus two surface. The
existence of such an action would imply the existence of a surface
subgroup, but it is not known, whether the existence of a surface
subgroup implies the existence of a periodic apartment. These groups
are the first candidates for groups that have no surface subgroups
arising from periodic apartments.

This talk is continuation to Alina Vdovina's talk on July 24th, but
can be followed also independently.

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