19 research outputs found
Finite index rigidity of hyperbolic groups
We prove that the topological complexity of a finite index subgroup of a
hyperbolic group is linear in its index. This follows from a more general
result relating the size of the quotient of a free cocompact action of
hyperbolic group on a graph to the minimal number of cells in a simplicial
classifying space for the group. As a corollary we prove that any two
isomorphic finite-index subgroups of a non-elementary hyperbolic group have the
same index.Comment: 25 pages, 4 figure
Uniform symplicity of groups with proximal action
We prove that groups acting boundedly and order-primitively on linear orders
or acting extremly proximality on a Cantor set (the class including various
Higman-Thomson groups and Neretin groups of almost automorphisms of regular
trees, also called groups of spheromorphisms) are uniformly simple. Explicit
bounds are provided.Comment: 23 pages, appendix by Nir Lazarovich, corrected versio
CAT(0) Polygonal Complexes are 2-Median
Median spaces are spaces in which for every three points the three intervals
between them intersect at a single point. It is well known that rank-1 affine
buildings are median spaces, but by a result of Haettel, higher rank buildings
are not even coarse median. We define the notion of ``2-median space'', which
roughly says that for every four points the minimal discs filling the four
geodesic triangles they span intersect in a point or a geodesic segment. We
show that CAT(0) Euclidean polygonal complexes, and in particular rank-2 affine
buildings, are 2-median. In the appendix, we recover a special case of a result
of Stadler of a Fary-Milnor type theorem and show in elementary tools that a
minimal disc filling a geodesic triangle is injective.Comment: 42 pages, 30 figures; final version - published in Geometriae
Dedicat