In this paper the author introduces a new subgroup In\((G)\) for a group \(G\) of finite Morley rank. He calls a uniformly definable family \(\mathcal F\) of connected subgroups geometric if a generic element of \(G\) lies in a unique element of \(\mathcal F\). An element \(g\in G\) is geometric if there is a geometric family which avoids \(g\), otherwise it is non-geometric or inevitable. The set of inevitable elements of \(G\) is called In\((G)\); the group \(G\) is geometric if every non-trivial element is geometric. Then In\((G)\) is a definable, definably characteristic subgroup which behaves well under quotients, and \(G/\text{In}(G)\) is geometric. If \(G_n\) is the direct product of \(n\) copies of \(G\) and \(\rho_n\) is the inclusion \(g\mapsto(g,1,\ldots,1)\), we define In\(_P(G)=\bigcap_n\rho_n^{-1}(\text{In}(G_n))\). Then \(G/\text{In}_P(G)\) injects definably into a geometric group. The author conjectures that a geometric group of finite Morley rank is definably linear over a ring \(K_1\oplus\cdots\oplus K_n\) of interpretable fields.
The author then studies In\(_P(G)\) for an algebraic group \(G\), and conjectures that it should be central. For a connected group of finite Morley rank he conjecturs that In\(_P(G)\) should be hypercentral, and shows that this follows from the preceding (main) conjecture. In the last section, the author proves that a connected geometric group of Morley rank 2 either has finite exponent or is isomorphic to \(K^+\ltimes K^\times\) or to \(K^+\oplus K^+\) for some interpretable field \(K\).
The notion of inevitable subgroup generalizes various known concepts (hypercentre, intersection of the Carter or Borel subgroups) and should turn out to be very useful in the study of non-simple groups of finite Morley rank. Moreover, it should also generalise to the wider context of stable groups.