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:
“You must train the children to their studies in a playful manner, and without any air of constraint, with the further object of discerning more readily the natural bent of their respective characters.”
—Plato (c. 427–347 B.C.)
“He was good-natured to a degree of weakness, even to tears, upon the slightest occasions. Exceedingly timorous, both personally and politically, dreading the least innovation, and keeping, with a scrupulous timidity, in the beaten track of business as having the safest bottom.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)
“When I had mapped the pond ... I laid a rule on the map lengthwise, and then breadthwise, and found, to my surprise, that the line of greatest length intersected the line of greatest breadth exactly at the point of greatest depth.”
—Henry David Thoreau (1817–1862)