Deciding finiteness for matrix groups over function fields |
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Authors: | Daniel N Rockmore Ki-Seng Tan Robert Beals |
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Institution: | (1) Department of Mathematics, Dartmouth College, 03755 Hanover, NH, USA;(2) Department of Mathematics, National Taiwan University, 106 Taipei, Taiwan;(3) Department of Mathematics, University of Arizona, 85721 Tuscon, AZ, USA |
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Abstract: | LetF be a field andt an indeterminate. In this paper we consider aspects of the problem of deciding if a finitely generated subgroup of GL(n,F(t)) is finite. WhenF is a number field, the analysis may be easily reduced to deciding finiteness for subgroups of GL(n,F), for which the results of 1] can be applied. WhenF is a finite field, the situation is more subtle. In this case our main results are a structure theorem generalizing a theorem
of Weil and upper bounds on the size of a finite subgroup generated by a fixed number of generators with examples of constructions
almost achieving the bounds. We use these results to then give exponential deterministic algorithms for deciding finiteness
as well as some preliminary results towards more efficient randomized algorithms.
Supported in part by NSF DMS Awards 9404275 and Presidential Faculty Fellowship. |
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