34 research outputs found
Covering R-trees, R-free groups, and dendrites
We prove that every length space X is the orbit space (with the quotient
metric) of an R-tree T via a free action of a locally free subgroup G(X) of
isometries of X. The mapping f:T->X is a kind of generalized covering map
called a URL-map and is universal among URL-maps onto X. T is the unique R-tree
admitting a URL-map onto X. When X is a complete Riemannian manifold M of
dimension n>1, the Menger sponge, the Sierpin'ski carpet or gasket, T is
isometric to the so-called "universal" R-tree A_{c}, which has valency equal to
the cardinality of the continuum at each point. In these cases, and when X is
the Hawaiian earring H, the action of G(X) on T gives examples in addition to
those of Dunwoody and Zastrow that negatively answer a question of J. W. Morgan
about group actions on R-trees. Indeed, for one length metric on H, we obtain
precisely Zastrow's example.Comment: This paper is the result of splitting off some of the results in the
preprint "Covering R-trees" and adding additional applications to R-free
group
Perfect and almost perfect homogeneous polytopes
The paper is devoted to perfect and almost perfect homogeneous polytopes in
Euclidean spaces. We classified perfect and almost perfect polytopes among all
regular polytopes and all semiregular polytopes excepting Archimedean solids
and two four-dimensional Gosset polytopes. Also we construct some non-regular
homogeneous polytopes that are (or are not) perfect and posed some unsolved
questions.Comment: 18 pages, 2 figure
Isometries, submetries and distance coordinates on Finsler manifolds
This paper considers fundamental issues related to Finslerian iso-
metries, submetries, distance and geodesics. It is shown that at each
point of a Finsler manifold there is a distance coordinate system. Us-
ing distance coordinates, a simple proof is given for the Finslerian
version of the Myers-Steenrod theorem and for the differentiability of
Finslerian submetries