Dual automorphisms of free groups
Fedaa Ibrahim, Martin Lustig

TL;DR
This paper introduces dual automorphisms of free groups, providing an algorithm to determine how automorphisms act on cylinders in the boundary of free groups, with bounds related to Nielsen automorphisms.
Contribution
It defines dual automorphisms based on cylinders, proves bounds on their complexity, and offers an efficient algorithm for their computation.
Findings
At most 2N distinct sets U_i describe automorphism action on cylinders.
Each U_i's size is bounded by 2^t, with t related to Nielsen automorphisms.
The dual automorphism depends only on the last letter of the word.
Abstract
For any choice of a basis the free group of finite rank can be canonically identified with the set of reduced words in . However, such a word admits a second interpretation, namely as cylinder . The subset of defined by depends not only on the element of given by the word , but also on the chosen basis . In particular one has in general, for : Indeed, the image of a cylinder under an automorphism is in general not a cylinder, but a finite union of cylinders: In his thesis the first author has given an efficient algorithm and a formula how to determine such a (uniquely determined) finite {\em reduced} set $U =…
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Taxonomy
TopicsGeometric and Algebraic Topology · semigroups and automata theory · Finite Group Theory Research
