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# Algebraic Graph Theory International Webinar (15.2.2022)

## v utorok 15.2.2022 o 19:00 hod.

14. 02. 2022 09.31 hod.

The organizers of the Algebraic Graph Theory International Webinar would like to invite you to join us and other colleagues on Februar 15, 2022, at 7pm Central European Summer Time (= 6pm UTC), for the next presentation delivered by Gareth A. Jones.

The title of Martin's talk: Maps and maximal subgroups

Abstract:
In 1933 Bernhard Neumann used permutations to construct uncountably many conjugacy classes of subgroups of ${\rm SL}_2(\mathbb Z)$ which act regularly on the primitive elements of $\mathbb Z^2$. Their images in the modular group $\Gamma={\rm PSL}_2(\mathbb Z)\cong {\rm C}_3*{\rm C}_2$ are maximal nonparabolic subgroups, i.e.~maximal with respect to containing no elements with a single fixed point in ${\mathbb P}^1({\mathbb Q})$. Further examples of such subgroups were later found by Magnus, Tretkoff, and Brenner and Lyndon.
I shall use planar maps to strengthen and extend their results by constructing uncountably many conjugacy classes of subgroups of $\Gamma$ which are maximal (as subgroups of $\Gamma$) {\sl and\/} nonparabolic. For example, all except one of Neumann's classes of subgroups are maximal. These subgroups all act regularly on the Farey graph, so this is a Cayley graph for all of them.
A similar construction gives, for all $p\ge 3$, $q\ge 2$, uncountably many conjugacy classes of subgroups of the triangle group $\Delta(p,q,\infty)\cong {\rm C}_p*{\rm C}_q$ which are maximal and nonparabolic. As a consequence, given any integer $p\ge 3$, every countable group is isomorphic to the automorphism group of uncountably many non-isomorphic $p$-valent maps on surfaces. I shall give evidence to support conjectures that every non-elementary Fuchsian group has uncountably many conjugacy classes of maximal subgroups, and hence that a similar realisation result holds for hypermaps of any given hyperbolic type.

The Zoom link for this semester is:
https://cuaieed-unam.zoom.us/j/87193320713?pwd=cHpiWUtYWlUvWHZjdGZteSt1QmZ5UT09
Meeting ID: 871 9332 0713
Passcode: 653250

Further details may be found at http://euler.doa.fmph.uniba.sk/AGTIW.html

where you can also find the slides and the recordings of our previous presentations. Also, if you wish to advertise an AGT friendly conference on this page, please send us the link.

Hoping to see you at the webinar, and wishing you all the best.