Burnside-type problems in discrete geometry

The paper is concerned with systems of incidence involving a space of points $X$ and lines consisting of $q$ points each. A free space $X$ is defined. For a space $X$ an analogue of the Burnside problem (solved in the negative) and an analogue of the weakened Burnside problem are formulated. In the case $q=3$ the positive answer to the analogue of the weakened Burnside problem is equivalent to the existence of a universal finite geometry.