A January 2005 invitation to random groups

By Yann Ollivier. Ensaios matemáticos, 2005

135 citations recorded by the source corpus.

Random groups provide a rigorous way to tackle such questions as “What does a typical (finitely generated) group look like?” or “What is the behavior of an element of a group when nothing particular happens?” We review the results obtained on random groups as of January 2005. We give proper definitions and list known properties of typical groups. We also emphasize properties of random elements in a given group. In addition we present more specific, randomly twisted group constructions providing new, “wild” examples of groups. A comprehensive discussion of open problems and perspectives is included. 2000 Mathematics Subject Classification: 20F65, 20P05, 60B99, 20F05, 20F67.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/309378696156cbc58bc52c185129c67f8956efc8

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Discrete Mathematics and Combinatorics

1. D. Fisher
2. Matthew Kahle
3. Elizabeth Meckes

In Geometry and Topology

1. D. Fisher
2. Matthew Kahle
3. Michael Farber
4. Armindo Costa
5. D. Yogeshwaran
6. Robert J. Adler

In Mathematical Physics

1. D. Fisher
2. Matthew Kahle
3. Michael Farber
4. Armindo Costa
5. Robert J. Adler
6. D. Yogeshwaran
7. Elizabeth Meckes

Citations in the corpus

• Groups acting on manifolds: around the Zimmer program (2008, background): “s on compact manifolds. It seems quite likely that the conjecture could be true for random groups in the density model with density more than 1 3 . These groups have property (T), see Ollivier’s book [186] for discussion on random groups. For the conjecture to be literally true would require that a ”
• Cubulating random groups at density less than 1/6 (2011, unknown): “The interest of the model is established through the following sharp phase transition theorem, proven by Gromov [Gro93] (see also [Oll04]): One of the motivations for the results in this paper is the following ([ ˙Zuk03]; see also the discussion in [Oll05b]): Theorem 1.3 (A. ˙Zuk) .”
• Topology of random simplicial complexes: a survey (2013, background): “This turns out to be true under a variety of different measures — see Ollivier’s survey [44].”