#### Alladi Ramakrishnan Hall

#### The BNS-invariant and the twisted conjugacy.

#### Parameswaran Sankaran

##### CMI

*Let G be a finitely generated group. The Bieri-Neumann-Strebel invariant \Sigma(G) is a subset of its character sphere S(G). The character sphere of G is the space of rays (emanating from the origin) in the vector space of all homomorphisms from G to the reals. *

An automorphism f of a group G defines, in a natural manner, an equivalence relation on G, namely the f-twisted conjugacy. This generalizes the notion of the usual conjugacy relation when the automorphism is the identity. The number of f-twisted conjugacy classes is called the Reidemeister number of f.

The group G is said to be Reidemeister-infinite (or R-infinity group) if the Reidemeister number is infinite for every automorphism of G.

We will define the BNS- invariant and illustrate it with examples. We will explain how it can be used in determining, in certain favourable situations, whether a group G is Reidemeister-infinite. We will illustrate this technique for certain groups of PL homeomorphisms of the interval, based on joint work with D L GonÃ§alves and R Strebel.

Done