arXiv subjects

Amy Tao

Publications and source records attributed to Amy Tao.

3 recordsLinked to original sources

arXiv2026

Commensurability and quasi-isometry classification for one vertex one loop tubular groups

A tubular group has a graph of groups decomposition with $\mathbb{Z}^2$ vertex groups and $\mathbb{Z}$ edge groups. This paper gives a classification of one vertex one loop tubular groups $G_{(k,\ell),(m,n)} = \langle a,b,t: [a,b]=1, t a^m b^n t^{-1} = a^k b^\ell \rangle$ up to commensurability and quasi-isometry. We show that tubular groups where the images of the edge maps have nonzero ``intersection number" are all commensurable and so quasi-isometric. When the intersection number is zero, there are two quasi-isometry classes and infinitely many commensurability classes. The nonzero and zero intersection number tubular groups are not quasi-isometric.

math.GR↗

arXiv2023

Friezes over $\mathbb Z[\sqrt{2}]$

A frieze on a polygon is a map from the diagonals of the polygon to an integral domain which respects the Ptolemy relation. Conway and Coxeter previously studied positive friezes over $\mathbb{Z}$ and showed that they are in bijection with triangulations of a polygon. We extend their work by studying friezes over $\mathbb Z[\sqrt{2}]$ and their relationships to dissections of polygons. We largely focus on the characterization of unitary friezes that arise from dissecting a polygon into triangles and quadrilaterals. We identify a family of dissections that give rise to unitary friezes and conjecture that this gives a complete classification of dissections which admit a unitary frieze.

math.CO↗

arXiv2021

Topology of augmented Bergman complexes

The augmented Bergman complex of a matroid is a simplicial complex introduced recently in work of Braden, Huh, Matherne, Proudfoot and Wang. It may be viewed as a hybrid of two well-studied pure shellable simplicial complexes associated to matroids: the independent set complex and Bergman complex. It is shown here that the augmented Bergman complex is also shellable, via two different families of shelling orders. Furthermore, comparing the description of its homotopy type induced from the two shellings re-interprets a known convolution formula counting bases of the matroid. The representation of the automorphism group of the matroid on the homology of the augmented Bergman complex turns out to have a surprisingly simple description. This last fact is generalized to closures beyond those coming from a matroid.

math.CO↗