In the mathematical subject of geometric group theory a train track map is a continuous map f from a finite connected graph to itself which is a homotopy equivalence and which has particularly nice cancellation properties with respect to iterations. This map sends vertices to vertices and edges to nontrivial edge-paths with the property that for every edge e of the graph and for every positive integer n the path fn(e) is immersed, that is fn(e) is locally injective on e. Train-track maps are a key tool in analyzing the dynamics of automorphisms of finitely generated free groups and in the study of the Culler–Vogtmann Outer space.
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Famous quotes containing the words train, track and/or map:
“Everything one does in life, even love, occurs in an express train racing toward death. To smoke opium is to get out of the train while it is still moving. It is to concern oneself with something other than life or death.”
—Jean Cocteau (1889–1963)
“To most men, experience is like the stern lights of a ship, which illumine only the track it has passed.”
—Samuel Taylor Coleridge (1772–1834)
“Unless, governor, teacher inspector, visitor,
This map becomes their window and these windows
That open on their lives like crouching tombs
Break, O break open,”
—Stephen Spender (1909–1995)