# Kropholler's Type of Tits Alternative for Generalised Baumslag-Solitar Groups.

In "Recent Results on Generalized Baumslag-Solitar Groups," by D.J.S Robinson in Note Mat. 30 (2010) suppl. n. 1, 37–53, it is claimed that

Kropholler [ . . . ] showed that there is a type of Tits alternative for GBS-groups, (Kropholler [11]).

[11] Baumslag-Solitar groups and some other groups of cohomological dimension two

The problem is that I need to know what type of Tits alternative holds for GBS-groups but, unfortunately, I can't get a hold of the relevant paper. It's behind a pay wall and my local university library isn't subscribed to the journal.

Please would someone clarify the type of Tits alternative for the GBS-groups for me here?

:)

Recent results on generalized Baumslag-Solitar groups does describe what they mean (maybe in a somewhat round about way).

The type of Tits alternative is this: Finitely generated subgroups of GBS-groups are either solvable or contain non-abelian free subgroups.

The above statement follows from the theorems they mention:

Theorem 1: The second derived subgroup of a GBS-group is free.

In particular GBS-groups are solvable exactly when the second derived subgroup is trivial or $$\mathbb Z$$. Otherwise it contains a higher rank free subgroup, so can not be solvable.

Theorem 2: Finitely generated subgroups of GBS-group are either free or GBS-groups.

This theorem gives us a way to apply theorem 1 to finitely generated subgroups of GBS-groups. This finishes the proof of "finitely generated Tits alternative".

• Well spotted. Thank you :) – Shaun Nov 19 '18 at 17:51