Maria Cumplido - Retractions to parabolic subgroups in Artin groups beyond the even case

Abstract

An Artin group is defined with a Presentation with a finite set of generators $S$ and, for each pair of generators $s,t\in S$, a single relation (or none) having the form $sts\cdots =tst\cdots$ where the number of letters on each side of the equality is the same and is denoted by $m_{s,t}$. Despite their simple definition, little is generally known about Artin groups (for example, the general solution to any Dehn problem is unknown). When the length $m_{s,t}$ of all existing relations is even, the Artin group is said to be even. Also, given an Artin group with its standard presentation, a standard parabolic subgroup is the subgroup generated by a subset of generators. In even Artin groups, there exists an obvious retraction to their standard parabolic subgroups that sends a generator to itself if it belongs to the subgroup and to $1$ otherwise. Recently, Antolín and Foniqi showed that these retractions are immensely useful for obtaining new results in even Artin groups. This joint work with Bruno A. Cisneros de la Cruz and Islam Foniqi aims to answer the natural question: what happens when we introduce odd-length relations? How can we define retractions, and how do they generalize the existing results?

Talk

We define Artin Groups and Coxeter Groups. Listening to the talks I get the impression that they are really useful in Europe.

Artin groups are mysterious! There are many properties we still don’t know! A standard subgroup is a special kind of subgroup. Conjugates of standard subgroups are called Parabolic subgroup.

This talk will only focus on some families of Artin groups.

• spherical?finite type (both are the same)\

Many groups are defined by graphs

• $A_n$: Graph in line form: Braid groups
• $B_n$: Line witha four at the end
• $D_n$: Line with a small $2$-fork at the end (see Luis Paris - Artin groups of type Dn)
• $E_6$ Fife line segments with a sixth splitting from the middle
• And much more

There is some theorem that Every complete subgraph with no $\infty$ defines a spherical Artin group (invisible $2$-edges are counted towards complete)

Cumplido shows how we can do retractions to $A_x$ if all the relator-powers are even. Antolin & Foruqi 2022, 2023 shows some thing on those retractions. Cumpido presents a use of retractions by simplifying a proof to one line (assuming a retraction exists).

Natural question: Can we generalize the result on retractions?

1. Find a family other than even admitting retractions of FC-type (whatever that is). Examples are given by triangular graphs with infinity edges and soem triangles without infinity edges. We classify the latter triangles into four groups, some of which do allow retractions and some which don’t. We get the theorem by Cisneros, Cumplido, Foruqi, namely: An FC-type Artin group admits retractions if and only if we have triangles of certain types. Moreover extra information on the subgroups is given.

Wenn ich eine Präsentation gebe, denke ich oft in Blackboxen. Eine gute Präsentation sollte mir daher diesen Prozess möglichst vereinfachen.

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