Tits alternative

In geometric group theory, a group G is said to satisfy the Tits alternative if for every subgroup H of G either H is virtually solvable or H contains a nonabelian free subgroup (in some versions of the definition this condition is only required to be satisfied for all finitely generated subgroups of G).

Examples of groups satisfying the Tits alternative which are either not linear, or at least not known to be linear, are:

• Hyperbolic groups
• Mapping class groups;^[3][4]
• Out(Fn);^[5]
• Certain groups of birational transformations of algebraic surfaces.^[6]

Examples of groups not satisfying the Tits alternative are:

• the Grigorchuk group;
• Thompson's group F.

The proof of the original Tits alternative^[1] is by looking at the Zariski closure of ${\displaystyle G}$ in ${\displaystyle \mathrm {GL} _{n}(k)}$. If it is solvable then the group is solvable. Otherwise one looks at the image of ${\displaystyle G}$ in the Levi component. If it is noncompact then a ping-pong argument finishes the proof. If it is compact then either all eigenvalues of elements in the image of ${\displaystyle G}$ are roots of unity and then the image is finite, or one can find an embedding of ${\displaystyle k}$ in which one can apply the ping-pong strategy.

Note that the proof of all generalisations above also rests on a ping-pong argument.