30 research outputs found
k-irreducible triangulations of 2-manifolds
This thesis deals with k-irreducible triangulations of closed, compact 2-manifolds without boundary. A triangulation is k-irreducible, if all its closed cycles of length less than k are nullhomotopic and no edge can be contracted without losing this property. k-irreducibility is a generalization of the well-known concept of irreducibility, and can be regarded as a measure of how closely the triangulation approximates a smooth version of the underlying surface.
Research follows three main questions: What are lower and upper bounds for the minimum and maximum size of a k-irreducible triangulation? What are the smallest and biggest explicitly constructible examples? Can one achieve complete classifications for specific 2-manifolds, and fixed k
Ultrametric properties for valuation spaces of normal surface singularities
Let be a fixed branch -- that is, an irreducible germ of curve -- on a
normal surface singularity . If are two other branches, define
, where
denotes the intersection number of and . Call arborescent if all the
dual graphs of its resolutions are trees. In a previous paper, the first three
authors extended a 1985 theorem of P{\l}oski by proving that whenever is
arborescent, the function is an ultrametric on the set of branches on
different from . In the present paper we prove that, conversely, if is
an ultrametric, then is arborescent. We also show that for any normal
surface singularity, one may find arbitrarily large sets of branches on ,
characterized uniquely in terms of the topology of the resolutions of their
sum, in restriction to which is still an ultrametric. Moreover, we
describe the associated tree in terms of the dual graphs of such resolutions.
Then we extend our setting by allowing to be an arbitrary semivaluation on
and by defining on a suitable space of semivaluations. We prove that
any such function is again an ultrametric if and only if is arborescent,
and without any restriction on we exhibit special subspaces of the space of
semivaluations in restriction to which is still an ultrametric.Comment: 50 pages, 14 figures. Final versio
A classification theorem for boundary 2-transitive automorphism groups of trees
Let be a locally finite tree all of whose vertices have valency at least
. We classify, up to isomorphism, the closed subgroups of
acting -transitively on the set of ends of and whose local action at
each vertex contains the alternating group. The outcome of the classification
for a fixed tree is a countable family of groups, all containing two
remarkable subgroups: a simple subgroup of index and (the semiregular
analog of) the universal locally alternating group of Burger-Mozes (with
possibly infinite index). We also provide an explicit example showing that the
statement of this classification fails for trees of smaller degree.Comment: 46 pages, 8 figure
AUTOMORPHISM GROUPS OF MAPS, SURFACES AND SMARANDACHE GEOMETRIES
Automorphism groups survey similarities on mathematical systems, which appear nearly in all mathematical branches, such as those of algebra, combinatorics, geometry, · · · and theoretical physics, theoretical chemistry, etc.. In geometry, configurations with high symmetry born symmetrical patterns, a kind of beautiful pictures in aesthetics. Naturally, automorphism groups enable one to distinguish systems by similarity. More automorphisms simply more symmetries of that system. This fact has established the fundamental role of automorphism groups in modern sciences. So it is important for graduate students knowing automorphism groups with applications