Regular left-orders on groups

A regular left-order on finitely generated group $G$ is a total, left-multiplication invariant order on $G$ whose corresponding positive cone is the image of a regular language over the generating set of the group under the evaluation map. We show that admitting regular left-orders is stable under extensions and wreath products and give a classification of the groups all whose left-orders are regular left-orders. In addition, we prove that solvable Baumslag-Solitar groups $B(1,n)$ admits a regular left-order if and only if $n\geq -1$. Finally, Hermiller and Sunic showed that no free product admits a regular left-order, however we show that if $A$ and $B$ are groups with regular left-orders, then $(A*B)\times \mathbb{Z}$ admits a regular left-order.Comment: 41 pages,9 figure

Cone types and geodesic languages for lamplighter groups and Thompson's group F

We study languages of geodesics in lamplighter groups and Thompson's group F. We show that the lamplighter groups $L_n$ have infinitely many cone types, have no regular geodesic languages, and have 1-counter, context-free and counter geodesic languages with respect to certain generating sets. We show that the full language of geodesics with respect to one generating set for the lamplighter group is not counter but is context-free, while with respect to another generating set the full language of geodesics is counter and context-free. In Thompson's group F with respect to the standard finite generating set, we show there are infinitely many cone types and no regular language of geodesics with respect to the standard finite generating set. We show that the existence of families of "seesaw" elements with respect to a given generating set in a finitely generated infinite group precludes a regular language of geodesics and guarantees infinitely many cone types with respect to that generating set.Comment: 30 pages, 13 figure

The total coordinate ring of a wonderful variety

We study the cone of effective divisors and the total coordinate ring of wonderful varieties, with applications to their automorphism group. We show that the total coordinate ring of any spherical variety is obtained from that of the associated wonderful variety by a base change of invariant subrings.Comment: Final version, to appear in Journal of Algebr

The Sasaki Join, Hamiltonian 2-forms, and Sasaki-Einstein Metrics

By combining the join construction from Sasakian geometry with the Hamiltonian 2-form construction from K\"ahler geometry, we recover Sasaki-Einstein metrics discovered by physicists. Our geometrical approach allows us to give an algorithm for computing the topology of these Sasaki-Einstein manifolds. In particular, we explicitly compute the cohomology rings for several cases of interest and give a formula for homotopy equivalence in one particular 7-dimensional case. We also show that our construction gives at least a two dimensional cone of both Sasaki-Ricci solitons and extremal Sasaki metrics.Comment: 38 pages, paragraph added to introduction and Proposition 4.1 added, Proposition 4.15 corrected, Remark 5.5 added, and explanation for irregular Sasaki-Einstein structures expanded. Reference adde