Disjoint circles: a classification

Abstract:

For q a prime-power, let ${\text {IP}}(q)$ denote the miquelian inversive plane of order q. The classification of certain translation planes of order ${q^2}$, called subregular, has been reduced to the classification of sets of disjoint circles in ${\text {IP}}(q)$. While R. H. Bruck has extensively studied triples of disjoint circles, this paper is concerned with sets of four or more circles in ${\text {IP}}(q)$. In a previous paper, the author has shown (for odd q) that the number of quadruples of disjoint circles in ${\text {IP}}(q)$ is asymptotic to ${q^{12}}/1536$. Hence a judicious approach to the classification problem is to study “interesting” quadruples. In general, let ${C_1}, \ldots ,{C_n}$ be a nonlinear set of n disjoint circles in ${\text {IP}}(q)$. Let H be the subgroup of the collineation group of ${\text {IP}}(q)$ composed of collineations that permute the ${C_i}$’s among themselves, and let K be that subgroup composed of collineations fixing each of the ${C_i}’s$. An interesting set of n disjoint circles would be one for which $K = 1$. It is shown that $K = 1$ if and only if \begin{equation}\tag {$*$} \left \{ {\begin {array}{*{20}{c}} {{\text {(i)}}\;{\text {there}}\;{\text {does}}\;{\text {not}}\;{\text {exist}}\;{\text {a}}\;{\text {circle}}\;D\;{\text {orthogonal}}\;{\text {to}}} \\ {{\text {each}}\;{\text {of}}\;{\text {the}}\;{\text {given}}\;n\;{\text {circles,}}\;{\text {and}}} \\ {{\text {(ii)}}\;{\text {we}}\;{\text {do}}\;{\text {not}}\;{\text {have}}\;{\text {one}}\;{\text {circle}}\;{\text {in}}\;{\text {our}}\;{\text {set}}\;{\text {orthogonal}}} \\ {{\text {to}}\;{\text {each}}\;{\text {of}}\;{\text {the}}\;{\text {other}}\;n - 1\;{\text {circles}}{\text {.}}} \\ \end{array} } \right .\end{equation} When $n = 4$ and under mild restrictions on q, an algorithm is developed that finds all nonlinear quadruples of disjoint circles satisfying the orthogonality conditions $( \ast )$ and having nontrivial group H. Given such a quadruple, the algorithm determines exactly what group H is acting. It is also shown that most quadruples in ${\text {IP}}(q)$, for large q, do indeed satisfy the conditions $( \ast )$. In addition, the cases when $n = 5,6,$ or 7 are explored to a lesser degree.