9,007 research outputs found
Generating self-map monoids of infinite sets
Let I be a countably infinite set, S = Sym(I) the group of permutations of I,
and E = End(I) the monoid of self-maps of I. Given two subgroups G, G' of S,
let us write G \approx_S G' if there exists a finite subset U of S such that
the groups generated by G \cup U and G' \cup U are equal. Bergman and Shelah
showed that the subgroups which are closed in the function topology on S fall
into exactly four equivalence classes with respect to \approx_S. Letting
\approx denote the obvious analog of \approx_S for submonoids of E, we prove an
analogous result for a certain class of submonoids of E, from which the theorem
for groups can be recovered. Along the way, we show that given two subgroups G,
G' of S which are closed in the function topology on S, we have G \approx_S G'
if and only if G \approx G' (as submonoids of E), and that cl_S (G) \approx
cl_E (G) for every subgroup G of S (where cl_S (G) denotes the closure of G in
the function topology in S and cl_E (G) its closure in the function topology in
E).Comment: 26 pages. In the second version several of the arguments have been
simplified, references to related literature have been added, and a few minor
errors have been correcte
Parabolic groups acting on one-dimensional compact spaces
Given a class of compact spaces, we ask which groups can be maximal parabolic
subgroups of a relatively hyperbolic group whose boundary is in the class. We
investigate the class of 1-dimensional connected boundaries. We get that any
non-torsion infinite f.g. group is a maximal parabolic subgroup of some
relatively hyperbolic group with connected one-dimensional boundary without
global cut point. For boundaries homeomorphic to a Sierpinski carpet or a
2-sphere, the only maximal parabolic subgroups allowed are virtual surface
groups (hyperbolic, or virtually ).Comment: 10 pages. Added a precision on local connectedness for Lemma 2.3,
thanks to B. Bowditc
The isotropy lattice of a lifted action
We obtain an algorithmic construction of the isotropy lattice for a lifted
action of a Lie group on and based only on the knowledge of
and its action on . Some applications to symplectic geometry are also shown.Comment: minor modification
The densest lattices in PGL3(Q2)
We find the smallest possible covolume for lattices in PGL3(Q2), show that
there are exactly two lattices with this covolume, and describe them
explicitly. They are commensurable, and one of them appeared in Mumford's
construction of his fake projective plane. We also discuss a new 2-adic
uniformization of another fake projective plane.Comment: Minor error correcte
The mass of unimodular lattices
The purpose of this paper is to show how to obtain the mass of a unimodular
lattice from the point of view of the Bruhat-Tits theory. This is achieved by
relating the local stabilizer of the lattice to a maximal parahoric subgroup of
the special orthogonal group, and appealing to an explicit mass formula for
parahoric subgroups developed by Gan, Hanke and Yu.
Of course, the exact mass formula for positive defined unimodular lattices is
well-known. Moreover, the exact formula for lattices of signature (1,n) (which
give rise to hyperbolic orbifolds) was obtained by Ratcliffe and Tschantz,
starting from the fundamental work of Siegel. Our approach works uniformly for
the lattices of arbitrary signature (r,s) and hopefully gives a more conceptual
way of deriving the above known results.Comment: 15 pages, to appear in J. Number Theor
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