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Effective Computations of Automatic Structures for HNN Extensions

Aurora DeBellevue, University of Nebraska - Lincoln


For a finitely presented group, the word problem asks if there is an algorithm which can always determine if a word over a generating set of the group represents the identity of the group. Boone and Novikov independently showed there exist finitely presented groups which do not have solvable word problem. Studying whether or not a finitely presented group has solvable word problem and constructing explicit algorithms which can solve the word problem is a rich and diverse field. Rewriting systems, automatic structures, and autostackable structures are specific algorithms to solve the word problem for some finitely presented groups. Generally, existence of some of these algorithms can depend on the presentation chosen; however for an automatic group, every finite presentation gives rise to an automatic structure. Research in this field includes searching for a word problem solution and studying the time or space complexity of the word problem solution.In this thesis we focus on automatic structures and the search for them using software. The software package kbmag is a part of GAP and can build an automatic structure for some finitely presented groups. The input to the program is a presentation, an ordering on the generating set, and an ordering to use when comparing words. There are three options in GAP and kbmag for the orderings, namely shortlex, weightlex, and wreath product. We investigate extending the capabilities of this software to fundamental groups of graphs of groups, and in particular HNN extensions of groups, which are known to be automatic, in some specific cases, when the vertex groups are automatic and each edge and adjacent vertex pair are strongly coset automatic.The mathematics, algorithms, and software developed for this thesis provide a method to produce automatic structures using the HNN structure, the automatic structure on the vertex groups, and the strongly coset automatic structures on each edge subgroup and vertex group pair. The program supported by the work in this dissertation is called AutGoG . Neither GAP nor kbmag currently calculate automatic structures using the graph of groups construction. We apply and extend the theorems proven in [12] to create explicit automatic structures of certain fundamental groups of graphs of groups.

Subject Area

Mathematics|Applied Mathematics

Recommended Citation

DeBellevue, Aurora, "Effective Computations of Automatic Structures for HNN Extensions" (2023). ETD collection for University of Nebraska-Lincoln. AAI30814247.