## Better term for a (simplicial) contractible plane continuum

In this joint paper that I should be working on, we make significant use of a certain generalization of a triangulated disk. Many of our important examples are triangulated disks, but we are also interested in certain simplicial complexes that are singular disks, or even more generally singular disks tiled by polygons. It is easy to describe the disks of interest to us as simplicial complexes which are contractible, compact subsets of the plane. The embedding in the plane only matters at all up to isotopy, and it also does not matter all that much; the most important condition is that the complex is contractible and at most two triangles meet at every edge. For instance, you can have two triangulated disks that meet at a vertex, trees, "barbells", etc.

One could more generally look at those topological spaces in the plane that are an intersection of nested, closed disks, or maybe those that are locally connected. For instance, the Mandelbrot set is one. We do not need them for what we are doing if they are not simplicial complexes, but this is an interesting class of topological spaces that should have a good name.

The following names have been proposed:

1. Van Kampen diagram - standard but ugly; very close to what we use although our edges are not labeled by group elements.
2. Contractible plane continuum - descriptive but clumsy.
3. Diskoid - a good name, by analogy with a dendroid which is the 1-dimensional case, but it doesn't seem to be standard.
4. Cactus - also a good name, by analogy with trees in graph theory, but it seems non-standard.
5. Singular disk - livable but not specific enough.

The reason that I want a short name is that the object X itself is used to make a moduli space or an algebraic variety. The moduli space is easy to describe: Assign fixed lengths to the edges of X and look at its rigid embeddings into a metric space. So we wouldn't want to say "contractible plane continuum variety". Also, for no particular reason I've been thinking of the triangulation as an extra decoration and instead name the underlying topological space.

I can certainly think of livable names, but the general idea appears in several places in mathematics and I would prefer a good name. Could someone with a good sense of the literature argue for a particular term, not necessarily one of the ones listed above? Or any useful opinion would be welcome.

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Cactus sounds nice. And you get to pluralize in -i, which always sounds cool... – Mariano Suárez-Alvarez Mar 1 2010 at 22:48
I don't have a good sense of the literature, but I'd argue for n-dendroid (2-dedroid in this case), the same way we call higher-dimensional knots "n-knots". It has the advantage of not cluttering the global name space. – Daniel Moskovich Mar 1 2010 at 23:20
"Cactus operad" is a fairly standard term for something a bit different from what you are describing. – Kevin Walker Mar 2 2010 at 0:40
As there seems to be no standard term, this is a good chance to invent a new one. Here's a candidate based on only a couple minutes' thought: polydendron. Sort of a variant on polyhedron bringing in the tree idea. A google search produces hits related to polymers, but nothing in math among the top items at least. (There are things called dendrimers, so that could be another possibility. Or just dendron, perhaps.) – Allen Hatcher Mar 2 2010 at 0:43
An answer based on a dream last night would be to call such an object a "tiger". Somehow, for me, at least last night in my sleep, a tiger is the image evoked for the object described. It's also unlikely to name-clash with other mathematical objects. – Daniel Moskovich Mar 4 2010 at 1:47

Prickly pear cactus is very pretty. Lots of cut points!

I believe that that your cacti are "tree-graded spaces" where are "pieces" are points or disks. See Drutu, C., Sapir, M.: Tree-graded spaces and asymptotic cones of groups. Topology 44, 959–1058 (2005).

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In the end, we (Joel Kamnitzer, his student Bruce Fontaine, and I) agreed on the term "diskoid". In the abstract, the word "cactus" seemed too clever by half. When I actually wrote it into the paper, it was awkward and it didn't emphasize the main property of interest, that our shapes are meant as a mild generalization of disks. Moreover, there have been papers that used the term "cactus" in similar but slightly different ways, and the mathematical etymology of this term is already somewhat confused. Actually, I don't have a good explanation of why the term that we picked seems good (or least bad), but somehow that is the feeling.

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