arXiv subjects

Brendon LaBuz

Publications and source records attributed to Brendon LaBuz.

7 recordsLinked to original sources

arXiv2021

Uniformly joinable, locally uniformly joinable, and weakly chained uniform spaces

Brodskiy, Dydak, LaBuz, and Mitra introduced the concepts of uniform joinability and local uniform joinability for uniform spaces when developing their theory of generalized uniform covering maps which was motivated by a paper by Berestovskii and Plaut. (Local) uniform joinability can be thought of as analogous to (local) path connectedness. A chain connected locally uniformly joinable uniform space is uniformly joinable. This note gives an example of a metric space that is uniformly joinable but not locally uniformly joinable. Plaut recently defined the concept of a weakly chained uniform space. We show that a weakly chained metrizable uniform space is locally uniformly joinable. Since local uniform joinability is equivalent to pointed 1-movability for metric continua, we find that weak chainability is equivalent to pointed 1-movability for such spaces.

math.GM↗

arXiv2020

Illustrating an error in "An equivalent condition for a uniform space to be coverable"

Berestovskii and Plaut introduced the concept of a coverable uniform space when developing their theory of generalized universal covering maps for uniform spaces. Brodskiy, Dydak, LaBuz, and Mitra introduced the concept of a locally uniformly joinable uniform space when developing their theory of generalized uniform covering maps which was motivated by the work of Berestovskii and Plaut. It is easy to see that a chain connected coverable uniform space is locally uniformly joinable. This paper points out an error in the attempt in Plaut's "An equivalent condition for a uniform space to be coverable" to prove that a locally uniformly joinable chain connected uniform space is coverable.

math.GT↗

arXiv2016

Big free groups acting on $Λ$-trees

The set of homotopy classes of based paths in the Hawaiian earring has a natural $\mathbb R$-tree structure, but under that metric the action by the fundamental group is not by isometries. Following a suggestion by Cannon and Conner, this paper defines an $\mathbb R^ω$-metric that does admit for an isometric action by the fundamental group. The space does not become an $\mathbb R^ω$-tree but is $0$-hyperbolic and embeds in an $\mathbb R^ω$-tree. Cannon and Conner define big free groups $\operatorname{BF}(c)$ for cardinal number $c$ which are a generalization of the fundamental group of the Hawaiian earring. They define a big Cayley graph which coincides with the set of homotopy classes of paths in the case of the Hawaiian earring. Instead of inserting real intervals to obtain the Cayley graph, we can insert $\mathbb R^c$-intervals and obtain a new $\mathbb R^c$-tree which admits an isometric action. In fact we do not need all of $\mathbb R^c$; we can insert $\mathbb Z^c$-intervals and obtain a $\mathbb Z^c$-tree. In the case of the Hawaiian earring we give a combinatorial description of the $\mathbb Z^ω$-tree and the corresponding action.

math.GR↗

arXiv2013

Sequential ends of metric spaces

We develop an analog to the ends of a metric space for the category of coarse metric spaces and show that it is equivalent to a previously defined coarse invariant.

math.MG↗

arXiv2011

A coarse invariant

This note extends the invariant defined in "An invariant of metric spaces under bornologous equivalences" to the coarse category.

math.MG↗

arXiv2011

A coarse invariant for all metric spaces

"An invariant of metric spaces under bornologous equivalences" gives an invariant and "A coarse invariant" extends the invariant to coarse equivalences. In both papers the invariant is defined for a class of metric spaces called sigma stable. This paper extends the invariant to all metric spaces and also gives an example of a space that is not sigma stable.

math.GN↗

arXiv2009

Inverse Limits of Uniform Covering Maps

In ``Rips complexes and covers in the uniform category'' the authors define, following James, covering maps of uniform spaces and introduce the concept of generalized uniform covering maps. Conditions for the existence of universal uniform covering maps and generalized uniform covering maps are given. This paper notes that the universal generalized uniform covering map is uniformly equivalent to the inverse limit of uniform covering maps and is therefore approximated by uniform covering maps. A characterization of generalized uniform covering maps that are approximated by uniform covering maps is provided as well as a characterization of generalized uniform covering maps that are uniformly equivalent to the inverse limit of uniform covering maps. Inverse limits of group actions that induce generalized uniform covering maps are also treated.

math.GN↗