Diagram groups and directed 2-complexes: Homotopy and homology

By Victor S. Guba, Mark V. Sapir. 2003

48 citations recorded by the source corpus.

We show that diagram groups can be viewed as fundamental groups of spaces of positive paths on directed 2-complexes (these spaces of paths turn out to be classifying spaces). Thus diagram groups are analogs of second homotopy groups, although diagram groups are as a rule non-Abelian. Part of the paper is a review of the previous results from this point of view. In particular, we show that the so-called rigidity of the R. Thompson's group F and some other groups is similar to the flat torus theorem. We find several finitely presented diagram groups (even of type F∞F∞) each of which contains all countable diagram groups. We show how to compute minimal presentations and homology groups of a large class of diagram groups. We show that the Poincare series of these groups are rational functions. We prove that all integer homology groups of all diagram groups are free Abelian.

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

View Paper: https://www.semanticscholar.org/paper/6fd62e84b18b78b4eb5b95e725be1662c59c8433

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Finiteness properties of cubulated groups (2012, background): “In particular, we note the work of Weinbaum using Dehn’s presentation to cubulate prime alternating link complements (see e.g., [Wis06]), the work of Aitchison–Rubinstein [AR90] cubulating certain 3–manifolds, the work of Brady–McCammond [McC09] who recognized how to thicken a complex to obtain a cu”
On Jones' subgroup of R. Thompson group $F$ (2015, background): “Such homomorphisms were called natural in [6].”
Metrics on diagram groups and uniform embeddings in a Hilbert space (2004, background): “It is proved in [15] that for any path q in T with the same endpoints as p there exists a unique reduced (p, q)-diagram over T .”
On subgroups of R. Thompson's group $F$ (2015, unknown): “To prove that H is proper we employ diagram groups of directed 2-complexes [14] and a 2-dimensional analog of Stallings foldings.”
Combinatorial algebra : syntax and semantics (2014, unknown): “The reason is that the area is so large that it requires a separate book (the survey by Kharlampovich and Sapir [177] is 250 pages long and does not even mention the most important results in the area obtained during the last 15 years).”