This book is the first volume in a two-volume set, which will
provide the complete proof of classification of two important
classes of geometries, closely related to each other: Petersen and
tilde geometries. There is an infinite family of tilde geometries
associated with non-split extensions of symplectic groups over a
field of two elements. Besides that there are twelve exceptional
Petersen and tilde geometries. These exceptional geometries are
related to sporadic simple groups, including the famous Monster
group and this volume gives a construction for each of the Petersen
and tilde geometries which provides an independent existence proof
for the corresponding automorphism group. Important applications of
Petersen and Tilde geometries are considered, including the
so-called Y-presentations for the Monster and related groups, and a
complete indentification of Y-groups is given. This is an essential
purchase for researchers into finite group theory, finite
geometries and algebraic combinatorics.
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