# Fall2017

All the talks are on Fridays, 4:00pm-5:00pm in room 5417 at the CUNY Graduate Center.

Oct 6. A. Miasnikov (Stevens) Equations: undecidability, decidability, complexity.

Abstract: We are going to talk about equations in groups and rings.

Oct 13. A. Vdovina (New Castle University and Hunter College CUNY) Expanders, surfaces and buildings

Abstract: We construct families of expanders as Cayley graphs of groups

with explicit presentations. These groups arise as quotients of groups acting on buildings.

We introduce a family of trivalent expanders which tessellate compact hyperbolic surfaces with large isometry groups.

We compare this family with Platonic graphs and prove topological and spectral properties of these families.

Oct 20. Z. Sunik (Hofstra U.) Ordering free groups, free products, and trees, not necessarily in that order

Abstract: We introduce left orders on the free group of any finite rank that are particularly easy to state and work with. The original approach orders uses actions of the free group on the real line, which is itself an ordered object, by letting the group just "borrow" the order from the line. We then recover the same orders, and a lot more, through a more organized and sophisticated approach. Namely, we order free products by first ordering their associated Bass-Serre trees and then letting the groups "borrow" the order from the trees. At the end, we discuss the complexity of the positive cones within the Chomsky hierarchy. Parts of the talk are based on joint works with Warren Dicks and with Susan Hermiller.

Oct. 27 L. Shneerson (Hunter College CUNY) Sharp bounds for the number of relators in irredundant presentations of inverse semigroups having polynomial growth.

Abstract: We give sharp lower bounds, in terms of a fixed number of generators, for the number of relators in irredundant presentations of Rees quotients of free inverse semigroups having polynomial growth. We also give a full description of all irredundant presentations that achieve this sharp bound. Some applications of these results will be discussed.

Joint work with David Easdown (University of Sydney, Australia).

Nov 3. B. Steinberg (CCNY) Homological finiteness properties for monoids acting on trees

Abstract: If a group G acts co-compactly on a tree with the vertex stabilizers of type FP_n and the edge stabilizers of type FP_{n-1}, then the group is of type FP_n. People have considered homological finiteness conditions for monoids for the past 30 years in connection with the problem of determining whether a monoid admits a finite complete rewriting system. Essentially all the work in this direction has been purely algebraic in nature and not geometric.

We consider monoid analogues of the above result for monoids acting on trees and as a consequence we are able to describe homological finiteness properties of special monoids in the sense of Adian and Makanin and of certain amalgamated free products and HNN extensions of monoids. In particular, we prove all special one-relator monoids are of type FP_{\infty}, providing a partial answer to a question of Kobayashi on homological finiteness properties of one-relator monoids.

This is joint work with Bob Gray (East Anglia).

Nov 10. S. Cleary (CCNY) Odd geometric behavior in Thompson's groups

Abstract: In the early 1960's Richard J. Thompson discovered a fascinating family of infinite groups in connection with his work in logic. These groups have reappeared in a wide variety of settings. Thompson's group F is the simplest known example of a variety of unusual group-theoretic phenomena and has been the subject of a great deal of study. I will describe these groups and their many generalizations from several different perspectives and discuss some of their remarkable properties, particularly some unusual aspects of the geometry of their standard Cayley graphs.

Nov. 17. (Joint with GRECS) E. Swenson (BYU) Wicks forms and normal forms for the mapping class group of a once punctured surface.

Joint with Alina Vdovina

Mosher obtains a automatic structure on the mapping class group of a once punctured surface of genus $g$ using ideal "triangulations" of the surface. We translate this into the setting of wicks forms of genus $g$ of maximal length, which I think of as a connected cubic graph $\Gamma$ on 4g-2 vertices with specified orientable circuit. Let $v$ be the vertex corresponding to $\Gamma$ in the Mosher complex $Y$. Any other vertex $w$ of $Y$ is uniquely represented as a finite sequence of paths (without backtracking) in $\Gamma$. Any edge path in $Y$ from $v$ to $w$ is realized as a sequence of elementary moves on $Y$ (each of which takes a path without backtracking to a path without backtracking). These moves will change each of the given paths into an empty path.

Nov 24. Thanksgiving

Dec 1. T. Koberda Embeddings of right-angled Artin groups

Abstract: We consider right-angled Artin subgroups of several classes of groups, concentrating on mapping class groups of surfaces, braid groups, and other right-angled Artin groups. We discuss the problem of deciding whether a particular right-angled Artin group embeds into the target group, and various applications to decision problems in group theory.

Most of this is joint work with Sang-hyun Kim.

Dec 8. Manhattan Algebra Day http://web.stevens.edu/algebraic/MADAY2017/