Singer cyclic lattices of type M

TL;DR

This paper constructs specific type-preserving lattices in buildings of type M with cyclic stabilizers, using amalgamations of generalized polygons and encoding their structure with gluing matrices.

Contribution

It introduces a method to construct and present Singer cyclic lattices of type M via amalgamation and gluing matrices, expanding understanding of these structures.

Findings

Constructed lattices for M with entries two, three, or infinity.

Established correspondence between gluing matrices and lattice presentations.

Demonstrated fundamental groups of quotient buildings as lattices.

Abstract

We construct the type-preserving panel-regular lattices in buildings of type M, for M with entries two, three or infinity, which have cyclic stabilizers of spherical 2-residues. We obtain these lattices as fundamental groups of their associated quotient buildings, which are constructed by amalgamating quotients of generalized digons and triangles by actions of Singer cyclic groups. Each quotient construction has an associated gluing matrix which encodes how the quotient generalized polygons are amalgamated. We show that these gluing matrices also encode presentations of the panel-regular lattices.