Panos Papasoglu
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View article: Graph Minors and Metric Spaces
Graph Minors and Metric Spaces Open
We present problems and results that combine graph-minors and coarse geometry. For example, we ask whether every geodesic metric space (or graph) without a fat H minor is quasi-isometric to a graph with no H minor, for an arbitrary finite …
View article: Minimal tetrahedra and an isoperimetric gap theorem in non-positive curvature
Minimal tetrahedra and an isoperimetric gap theorem in non-positive curvature Open
We investigate isoperimetric inequalities for Lipschitz 2-spheres in CAT(0) spaces, proving bounds on the volume of efficient null-homotopies. In one dimension lower, it is known that a quadratic inequality with a constant smaller than $c_…
View article: A coarse-geometry characterization of cacti
A coarse-geometry characterization of cacti Open
We give a quasi-isometric characterization of cacti, which is similar to Manning's characterization of quasi-trees by the bottleneck property. We also give another quasi-isometric characterization of cacti using fat theta curves.
View article: Graph minors and metric spaces
Graph minors and metric spaces Open
We present problems and results that combine graph-minors and coarse geometry. For example, we ask whether every geodesic metric space (or graph) without a fat $H$ minor is quasi-isometric to a graph with no $H$ minor, for an arbitrary fin…
View article: Periodic geodesics in singular spaces
Periodic geodesics in singular spaces Open
We extend the classical result of Lyusternik and Fet on the existence of closed geodesics to singular spaces. We show that if $X$ is a compact geodesic metric space satisfying the CAT($κ$) condition for some fixed $κ>0$ and $π_n(X)\ne 0$ f…
View article: Polynomial growth and asymptotic dimension
Polynomial growth and asymptotic dimension Open
Bonamy et al \cite{BBEGLPS} showed that graphs of polynomial growth have finite asymptotic dimension. We refine their result showing that a graph of polynomial growth strictly less than $n^{k+1}$ has asymptotic dimension at most $k$. As a …
View article: Asymptotic dimension of planes and planar graphs
Asymptotic dimension of planes and planar graphs Open
We show that the asymptotic dimension of a geodesic space that is homeomorphic to a subset in the plane is at most three. In particular, the asymptotic dimension of the plane and any planar graph is at most three.
View article: Uryson width and volume
Uryson width and volume Open
We give a short proof of a theorem of Guth relating volume of balls and Uryson width. The same approach applies to Hausdorff content implying a recent result of Liokumovich-Lishak-Nabutovsky-Rotman. We show also that for any $C>0$ there is…
View article: Finite cuts and CAT(0) boundaries
Finite cuts and CAT(0) boundaries Open
We show that if a 1-ended group $G$ acts geometrically on a CAT(0) space $X$ and $\bd X$ is separated by $m$ points then either $G$ is virtually a surface group or $G$ splits over a 2-ended group. In the course of the proof we study nestin…
View article: A surface with discontinuous isoperimetric profile
A surface with discontinuous isoperimetric profile Open
We show that there is a complete connected 2-dimensional Riemannian manifold with discontinuous isoperimetric profile, answering a question of Nardulli and Pansu.
View article: Short loops in surfaces with a circle boundary component
Short loops in surfaces with a circle boundary component Open
It is a classical theorem of Loewner that the systole of a Riemannian torus can be bounded in terms of its area. We answer a question of a similar flavor of Robert Young showing that if $T$ is a Riemannian 2-torus with boundary in $\mathbb…