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Showing that a group acts freely and discretely on real plane
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[QUOTE="lavinia, post: 6564748, member: 243745"] ##a## and ##b^2## generate a lattice with generators ##(0,1)## and ##(2,0)##. This is a free abelian group on two generators. Conjugating elements of the lattice by ##b## gives another point in the lattice so any element of the group can be written in the form ## a^{m}b^{2n}## or ##(a^{m}b^{2n})b## For instance ##ba=(bab^{−1})b## and ##(bab^{−1})## is an element of the lattice. So every element of the group is either a pure translation by a lattice point or a lattice translation following the action of ##b##. From this normal form, a direct calculation shows that the action is free. The group permutes the squares that are bounded by the straight lines parallel to the x and y axes and have integer intercepts. For instance, ##b## shifts the unit square to the adjacent square to the right along the x axis. Note that it does this without fixing any boundary points. Structurally, the group is an extension of a two dimensional lattice by the group with 2 elements. ##0→L^2→π→Z_2=π/L^2→0## The element ##b## is a lift of the generator of ##Z_2## to ##π##. Problem: Show that this group is torsion free i.e. it has no elements of finite order. Problem: Show that the two free abelian subgroups generated by ##a## and ##a^{-1}b## give another description of the group as a split extension of ##Z## by ##Z## ##0→Z→π↔Z→0## where the quotient ##Z## is generated by ##a^{-1}b## and the kernel is generated by ##a##. In technical language, ##π## is the semi-direct product of ##Z## with ##Z## in which the action of the quotient ##Z## on the kernel ##Z## by conjugation is multiplication by ##-1##. Question: Does this description also show that the group is torsion free? Problem: The action of ##b## on the lattice is the identity on ##(2,0)## and negation on ##(0,1)##. Suppose instead that the action was negation of both basis vectors. Would the group still act on the plane freely? Would the group still be torsion free? Same question if the action of ##b## is the identity on ##(2,0)## and maps ##(0,1)## to ##(2,-1)##. Problem: The quotient space of the plane by the action of the lattice generated by the translations ##(0,1)## and ##(2,0)## is a torus. Show that the action of ##b## on the plane projects to a properly discontinuous involution (map of order 2) of the torus. Problem: Construct a torsion free extension of the standard 3 dimensional lattice in ##R^3## by the group ##Z_2×Z_2## whose action by conjugation on the lattice is ##(x,y,z)→(-x,-y,z)## and ##(x,y,z)→(-x,y,-z)## and the product ##(x,-y,-z)## [/QUOTE]
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Showing that a group acts freely and discretely on real plane
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