12,447 research outputs found
The definability criterions for convex projective polyhedral reflection groups
Following Vinberg, we find the criterions for a subgroup generated by
reflections \Gamma \subset \SL^{\pm}(n+1,\mathbb{R}) and its finite-index
subgroups to be definable over where is an integrally
closed Noetherian ring in the field . We apply the criterions for
groups generated by reflections that act cocompactly on irreducible properly
convex open subdomains of the -dimensional projective sphere. This gives a
method for constructing injective group homomorphisms from such Coxeter groups
to \SL^{\pm}(n+1,\mathbb{Z}). Finally we provide some examples of
\SL^{\pm}(n+1,\mathbb{Z})-representations of such Coxeter groups. In
particular, we consider simplicial reflection groups that are isomorphic to
hyperbolic simplicial groups and classify all the conjugacy classes of the
reflection subgroups in \SL^{\pm}(n+1,\mathbb{R}) that are definable over
. These were known by Goldman, Benoist, and so on previously.Comment: 31 pages, 8 figure
From the hyperbolic 24-cell to the cuboctahedron
We describe a family of 4-dimensional hyperbolic orbifolds, constructed by
deforming an infinite volume orbifold obtained from the ideal, hyperbolic
24-cell by removing two walls. This family provides an infinite number of
infinitesimally rigid, infinite covolume, geometrically finite discrete
subgroups of the isometry group of hyperbolic 4-space. It also leads to finite
covolume Coxeter groups which are the homomorphic image of the group of
reflections in the hyperbolic 24-cell. The examples are constructed very
explicitly, both from an algebraic and a geometric point of view. The method
used can be viewed as a 4-dimensional, but infinite volume, analog of
3-dimensional hyperbolic Dehn filling.Comment: The article has 78 pages and 37 figures. Many of the figures use
color in an essential way. If possible, use a color printe
The Bianchi groups are subgroup separable on geometrically finite subgroups
We show that for certain arithmetic groups, geometrically finite subgroups
are the intersection of finite index subgroups containing them. Examples are
the Bianchi groups and the Seifert-Weber dodecahedral space. In particular, for
manifolds commensurable with these groups, immersed incompressible surfaces
lift to embeddings in a finite sheeted covering.Comment: 19 page
On the growth of cocompact hyperbolic Coxeter groups
For an arbitrary cocompact hyperbolic Coxeter group G with finite generator
set S and complete growth function P(x)/Q(x), we provide a recursion formula
for the coefficients of the denominator polynomial Q(x) which allows to
determine recursively the Taylor coefficients and the pole behavior of the
growth function of G in terms of its Coxeter subgroup structure. We illustrate
this in the easy case of compact right-angled hyperbolic n-polytopes. Finally,
we provide detailed insight into the case of Coxeter groups with at most 6
generators, acting cocompactly on hyperbolic 4-space, by considering the three
combinatorially different families discovered and classified by Lanner,
Kaplinskaya and Esselmann, respectively.Comment: 24 page
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