30 research outputs found
Splittings and the asymptotic topology of the lamplighter group
It is known that splittings of finitely presented groups over 2-ended groups
can be characterized geometrically. We show that this characterization does not
extend to all finitely generated groups. Answering a question of Kleiner we
show that the Cayley graph of the lamplighter group is coarsely separated by
quasi-circles.Comment: 11 page
Cheeger constants of surfaces and isoperimetric inequalities
We show that the Cheeger constant of compact surfaces is bounded by a
function of the area. We apply this to isoperimetric profiles of bounded genus
non-compact surfaces, to show that if their isoperimetric profile grows faster
than , then it grows at least as fast as a linear function. This
generalizes a result of Gromov for simply connected surfaces.
We study the isoperimetric problem in dimension 3. We show that if the
filling volume function in dimension 2 is Euclidean, while in dimension 3 is
sub-Euclidean and there is a such that minimizers in dimension 3 have genus
at most , then the filling function in dimension 3 is `almost' linear.Comment: 28 page
Group splittings and asymptotic topology
It is a consequence of the theorem of Stallings on groups with many ends that
splittings over finite groups are preserved by quasi-isometries. In this paper
we use asymptotic topology to show that group splittings are preserved by
quasi-isometries in many cases. Roughly speaking we show that splittings are
preserved under quasi-isometries when the vertex groups are fundamental groups
of aspherical manifolds (or more generally `coarse PD(n)-groups') and the edge
groups are `smaller' than the vertex groups.Comment: 14 page
Growth and isoperimetric profile of planar graphs
Let G be a planar graph such that the volume function of G satisfies V(2n)<
CV(n) for some constant C > 0. Then for every vertex v of G and integer n,
there is a domain \Omega such that B(v,n) \subset \Omega, \Omega \subset B(v,
6n) and the size of the boundary of \Omega is at most order n.Comment: 8 page
A surface with discontinuous isoperimetric profile and expander manifolds
We construct sequences of `expander manifolds' and we use them to show that
there is a complete connected 2-dimensional Riemannian manifold with
discontinuous isoperimetric profile, answering a question of Nardulli and
Pansu. Using expander manifolds in dimension 3 we show that for any there is a Riemannian 3-sphere of volume 1, such that any (not
necessarily connected) surface separating in two regions of volume greater
than , has area greater than .Comment: 15 pages, this paper merged with arXiv:1803.07375 (2018), to appear
in Geom. Dedicat
A cactus theorem for end cuts
Dinits-Karzanov-Lomonosov showed that it is possible to encode all minimal
edge cuts of a graph by a tree-like structure called a cactus. We show here
that minimal edge cuts separating ends of the graph rather than vertices can be
`encoded' also by a cactus. We apply our methods to finite graphs as well and
we show that several types of cuts can be encoded by cacti.Comment: 19 page
Simply connected homogeneous continua are not separated by arcs
We show that locally connected, simply connected homogeneous continua are not
separated by arcs. We ask several questions about homogeneous continua which
are inspired by analogous questions in geometric group theory.Comment: 15 pages, 8 figure
JSJ-decompositions of finitely presented groups and complexes of groups
A JSJ-splitting of a group over a certain class of subgroups is a graph
of groups decomposition of which describes all possible decompositions of
as an amalgamated product or an HNN extension over subgroups lying in the
given class. Such decompositions originated in 3-manifold topology. In this
paper we generalize the JSJ-splitting constructions of Sela, Rips-Sela and
Dunwoody-Sageev and we construct a JSJ-splitting for any finitely presented
group with respect to the class of all slender subgroups along which the group
splits. Our approach relies on Haefliger's theory of group actions on CAT
spaces