Abstract: Suppose $\cal S$ is a finite generalized quadrangle (GQ) of order $(s,t)$, $s\ne 1\ne t$, and suppose that $L$ is a line of $\cal S$. A symmetry about $L$ is an automorphism of the GQ which fixes every line of $\cal S$ meeting $L$ (including $L$). A line is called an axis of symmetry if there is a full group of symmetries of size $s$ about this line, and a point of a generalized quadrangle is a translation point if every line through it is an axis of symmetry. A GQ with a translation point is often called a translation generalized quadrangle. In the present paper, we classify the generalized quadrangles with at least two distinct translation points. In order to obtain the main result, we prove many more general theorems which are useful for the theory of span-symmetric generalized quadrangles (these are the GQ's with non-concurrent axes of symmetry), and using earlier results of the author, we give more general versions of our main theorem.
As a by-product of the proof of our main result, we will show that for any span-symmetric generalized quadrangle $\cal S$ of order $(s,t)$, $s\ne 1\ne t$, $s$ and $t$ are powers of the same prime, and if $s\ne t$ and $s$ is odd, then $\cal S$ always contains at least $s+1$ classical subquadrangles of order $s$.
In an addendum we obtain an explicit construction of some classes of spreads for the point-line duals of the Kantor flock generalized quadrangles as a second by-product of the proof of our main result.
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