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Canonical Hexagons and the $PSL(2,CC)$ Discreteness Problem | | Jane Gilman
; Linda Keen
; | | Date: |
2 Aug 2015 | | Abstract: | The discreteness problem, that is the problem of determining {sl whether or
not} a given finitely generated group $G$ of orientation preserving isometries
of hyperbolic three-space, $HH^3$ is discrete as a subgroup of the whole
isometry group $Isom(HH^3)$, is a challenging problem that has been
investigated for more than a century and is still open. It is known that $G$ is
discrete if, and only if, every non-elementary two generator subgroup is.
Several sufficient conditions for discreteness are also known as are some
necessary conditions, though no single necessary and sufficient condition is
known. There is a finite discreteness algorithm for the two generator subgroups
of $Isom(mathbb{H}^2)$. But the situation in $HH^3$ is more delicate because
there are geometrically infinite groups.
We present a {sl semi-algorithm} that is a procedure that terminates
sometimes but not always. There is no standard way to find an infinite sequence
of distinct elements that converges to the identity when a group is not
discrete. Our semi-algorithm either produces such an infinite sequence or finds
a finite sequence that produces a right angled hexagon in $HH^3$ which has a
special property that is a generalization of the notion of convexity. We call
it a {sl canonical hexagon}. If the group is discrete, free and geometrically
finite, it always has an essentially unique canonical hexagon which the
procedure finds in a finite number of steps. | | Source: | arXiv, 1508.0257 | | Services: | Forum | Review | PDF | Favorites | |
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