Speaker: Stefan Kohl (Stuttgart)
Title: What do Thompson's Group \(V\) and the Collatz Conjecture have in Common?
Abstract:
Actually let's start with a difference: to generate Thompson's group \(V\), one needs 4 class transpositions, while the Collatz conjecture can already be encoded with 3. -- Now, what is a class transposition?
Denote a residue class \(r+m\mathbb{Z}\) by \(r(m)\). Given disjoint residue classes \(r_1(m_1)\) and \(r_2(m_2)\) of the integers, the corresponding class transposition \(\tau_{r_1(m_1),r_2(m_2)}\) is the permutation of \(\mathbb{Z}\) which interchanges \(r_1 + km_1\) and \(r_2 + km_2\) for each integer \(k\) and which fixes all other points. Here it is understood that \(0 \leq r_i < m_i\).
With this we have \(V \cong \langle \tau_{0(2),1(2)}, \tau_{1(2),2(4)}, \tau_{0(2),1(4)}, \tau_{1(4),2(4)} \rangle\), where the four class transpositions correspond to the generators \(\kappa\), \(\lambda\), \(\mu\) and \(\nu\) already used by Higman in 1974.
On the other hand, the Collatz conjecture holds if and only if the group \(G := \langle \tau_{0(2),1(2)}, \tau_{1(2),2(4)}, \tau_{1(4),2(6)} \rangle\) acts transitively on \(\mathbb{N}_0\).
Apart from this, the talk will provide a brief survey of results on groups generated by few class transpositions obtained by means of computation.