| Date: | October 8, 2020 |
| Time: | Noon |
| Location: | online |
| Speaker: | Victor Guba, Vologda State University |
| Title: | On the Ore condition for the group ring of R. Thompson's group F |
| Abstract: | Let $R=K[G]$ be a group ring of a group $G$ over a field $K$. The Ore condition says that for any $a,b\in R$ there exist $s,t\in R$ such that $as=bt$ and $s\ne0$ or $t\ne0$. It always holds whenever $G$ is amenable. Recently it was shown that for R.\,Thompson's group $F$ the converse is also true. So the famous amenability problem for $F$ is equivalent to the question on the Ore condition for the same group.
It is easy to see that the problem on the Ore condition is equivalent to the same question for the monoid ring $K[M]$, where $M$ is the monoid of positive elements of $F$. In this paper we reduce the problem for the case when $a$, $b$ are homogeneous elements of the same degree in the monoid ring. We study the case of degree $1$ and find minimal solutions of the Ore equation. For the case of degree $2$, we study the case of linear combinations of monomials from $S=\{x_0^2,x_0x_1,x_0x_2,x_1^2,x_1x_2\}$. We show that this set is not doubling, that is, there are nonempty finite subsets $X\subset M\subset F$ such that $|SX| < 2|X|$. In particular, Ore condition holds for linear combinations of these monomials.
The case of monomials of higher degree is open as well as the case of degree $2$ for the monomials on $x_0,x_1,...,x_m$, where $m\ge3$. Recall that negative answer for one of these cases will immediately imply non-amenability of $F$. |
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| Date: | November 4, 2020 |
| Time: | Noon |
| Location: | online |
| Speaker: | George Willis, The University of Newcastle, Australia |
| Title: | Scale Groups |
| Abstract: | Scale groups are closed, vertex-transitive groups of
automorphisms of a regular tree that fix an end of the tree. These
concrete groups emerge from the structure theory of abstract totally
disconnected, locally compact groups.
There is also a very close correspondence between scale groups and
closed self-replicating groups, which are groups of automorphisms of a
rooted tree.
The role of scale groups in the study of general totally disconnected,
locally compact groups and their connection with self-replicating
groups will be explained in the talk. |
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| Date: | November 25, 2020 |
| Time: | Noon |
| Location: | online |
| Speaker: | Ignat Soroko, Louisiana State University |
| Title: | Groups of type FP: their quasi-isometry classes and homological Dehn functions |
| Abstract: | There are only countably many isomorphism classes of finitely presented groups, i.e. groups of type $F_2$. Considering a homological analog of finite presentability, we get the class of groups $FP_2$. Ian Leary proved that there are uncountably many isomorphism classes of groups of type $FP_2$ (and even of finer class FP). R.Kropholler, Leary and I proved that there are uncountably many classes of groups of type FP even up to quasi-isometries. Since `almost all' of these groups are infinitely presented, the usual Dehn function makes no sense for them, but the homological Dehn function is well-defined. In an on-going project with N.Brady, R.Kropholler and myself, we show that for any integer $k\ge4$ there exist uncountably many quasi-isometry classes of groups of type FP with a homological Dehn function $n^k$. In this talk I will give the relevant definitions and describe the construction of these groups. Time permitting, I will describe the connection of these groups to the Relation Gap Problem. |
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| Date: | December 2, 2020 |
| Time: | Noon |
| Location: | online |
| Speaker: | Wenhao Wang, Vanderbilt University |
| Title: | Dehn Functions of Finitely Presented Metabelian Groups |
| Abstract: | The Dehn function was introduced by computer scientists Madlener and Otto to describe the complexity of the word problem of a group, and also by Gromov as a geometric invariant of finitely presented groups. In this talk, I will show that the upper bound of the Dehn function of finitely presented metabelian group $G$ is $2^\{n^\{2k\}\}$, where $k$ is the minimal torsion-free rank of abelian group T such that there exists an abelian group $A$ satisfying $G/A \cong T$, answering the question that if the Dehn functions of metabelian groups are uniformly bounded. I will also talk about the relative Dehn function of finitely generated metabelian group and its relation to the Dehn function.
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