Fetching the paper…

Reading the bibliography…

The fundamental group of random 2-complexes · Around

2007

The fundamental group of random 2-complexes

Babson, Eric, Hoffman, Christopher, Kahle, Matthew

Original
PDF
Post on X

Understand

We study Linial-Meshulam random 2-complexes, which are two-dimensional analogues of Erd\H{o}s-R\'enyi random graphs.

We find the threshold for simple connectivity to be p = n^{-1/2}.
This is in contrast to the threshold for vanishing of the first homology group, which was shown earlier by Linial and Meshulam to be p = 2 log(n)/n.
We use a variant of Gromov's local-to-global theorem for linear isoperimetric inequalities to show that when p = O(n^{-1/2 -\epsilon}) the fundamental group is word hyperbolic.

Built on

On random graphs. I
P. Erdős and A. Rényi · 1959
Earlier work this paper cites.
Hyperbolic groups
M. Gromov · 1987
Earlier work this paper cites.
Asymptotic invariants of infinite groups
M. Gromov · 1993
Earlier work this paper cites.
A short proof that a subquadratic isoperimetric inequality implies a linear one
B. H. Bowditch · 1995
Earlier work this paper cites.
An algorithm detecting hyperbolicity
P. Papasoglu · 1996
Earlier work this paper cites.

Similar

Random graphs
Béla Bollobás · 2001
Cited alongside, same era.
Property (T) and Kazhdan constants for discrete groups
Original
A. Żuk · 2003
Cited alongside, same era.
A January 2005 invitation to random groups
Yann Ollivier · 2005
Cited alongside, same era.
Homological connectivity of random 2-complexes
Nathan Linial and Roy Meshulam · 2006
Cited alongside, same era.

Then

Topological characteristics of random triangulated surfaces
Nicholas Pippenger and Kristin Schleich · 2006
Later among the works it cites.
The neighborhood complex of a random graph
Matthew Kahle · 2007
Closest in time.
Topology of random clique complexes
Matthew Kahle · 2009
Closest in time.
Homological connectivity of random k k -dimensional complexes
R. Meshulam and N. Wallach · 2009
Closest in time.

Beyond the bibliography

alphaXiv searches the wider corpus for related work and actual follow-ups.

Open on alphaXiv

alphaXiv is searching for related work…