Automorphism groups of hyperbolic lattices. Based on the concept of dual cones introduced by J. Opgenorth [Exp. Math. 10, No.4, 599–608 (2001; Zbl 1007.20046)] we give an algorithm to compute a generating system of the group of automorphisms of an integral lattice endowed with a hyperbolic bilinear form. The paper will be organized as follows: In Section 2 we recall the basic definitions and key results about dual cones from the paper cited above which give a general method to determine generating systems of discontinuous groups acting on dual cones. The application of the results in Section 2 on hyperbolic lattices as well as a quite powerful way to shorten the calculation time is given in Section 3. In Section 4 we analyse the scope and running time of our algorithm and give some examples. These were calculated using the computer algebra system Magma. The source code for the necessary Magma-package AutHyp.m as well as a short description of the included intrinsics is available via the author’s homepage mmertens.

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  1. Mertens, Michael H.: Automorphism groups of hyperbolic lattices (2014)