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.