Infinitely Generated Braid Groups
There are many ways to generalize this notion to an infinite number of strands. The simplest way is take the direct limit of braid groups, where the attaching maps send the generators of to the first generators of (i.e., by attaching a trivial strand). Fabel has shown that there are two topologies that can be imposed on the resulting group each of whose completion yields a different group. One is a very tame group and is isomorphic to the mapping class group of the infinitely punctured disk — a discrete set of punctures limiting to the boundary of the disk.
The second group can be thought of the same as with finite braid groups. Place a strand at each of the points and the set of all braids — where a braid is defined to be a collection of paths from the points to the points so that the function yields a permutation on endpoints — is isomorphic to this wilder group. An interesting fact is that the pure braid group in this group is isomorphic to both the inverse limit of finite pure braid groups and to the fundamental group of the Hilbert cube minus the set .
Read more about this topic: Braid Group
Famous quotes containing the words infinitely, generated, braid and/or groups:
“It’s impossible to represent a saint [in Art]. It becomes boring. Perhaps because he is, like the Saturday Evening Post people, in the position of having almost infinitely free will.”
—W.H. (Wystan Hugh)
“It is precisely the purpose of the public opinion generated by the press to make the public incapable of judging, to insinuate into it the attitude of someone irresponsible, uninformed.”
—Walter Benjamin (1892–1940)
“As a father I had some trouble finding the words to separate the person from the deed. Usually, when one of my sons broke the rules or a window, I was too angry to speak calmly and objectively. My own solution was to express my feelings, but in an exaggerated, humorous way: “You do that again and you will be grounded so long they will call you Rip Van Winkle II,” or “If I hear that word again, I’m going to braid your tongue.””
—David Elkind (20th century)
“Trees appeared in groups and singly, revolving coolly and blandly, displaying the latest fashions. The blue dampness of a ravine. A memory of love, disguised as a meadow. Wispy clouds—the greyhounds of heaven.”
—Vladimir Nabokov (1899–1977)