arXiv subjects
Publications and source records attributed to Lucy Koch-Hyde.
We will answer a problem of Dotsenko that asks for a tight upper bound on the rank of subgroups given a maximum length for its generators. Then, we will use the same tools to answer a longstanding problem of Grigorchuk in the 10th edition of the Kourovka notebook. We accomplish this using Stallings foldings and simple graph theory.
We say a word in a free group is ``partially positive'' if there is at least one generator which appears in the word, but does not appear with negative exponent. We show that one-relator groups with a partially positive relator are residually solvable. This extends a theorem of Baumslag showing one-relator groups with positive relators are residually solvable.
A word in a free group is called ``potentially positive'' if it is automorphic to an element which is written with only positive exponents. We will develop automata to analyze properties of potentially positive words. We will use these to give new bounds on the asymptotic growth of potentially positive elements in free groups of 2 to 7 generators. We prove the bounds for $F_2$ are tight, giving the growth function up to a constant multiplier. We use the same tools to show that certain restricted automata cannot recognize the set of potentially positive elements.
By strengthening known results about primitivity-blocking words in free groups, we prove that for any nontrivial element w of a free group of finite rank, there are words that cannot be subwords of any cyclically reduced automorphic image of w. This has implications for the average-case complexity of a variant of Whitehead's problem.