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- W2922456536 abstract "In cite{KSS06} it was shown that with respect to the simple non-backtracking walk on the free group $F_N=F(a_1,dots,a_N)$ the Whitehead algorithm has strongly linear time generic-case complexity and that generic elements of $F_N$ are strictly in their $Out(F_N)$-orbits. Here we generalize these results, with appropriate modifications, to a much wider class of processes generating elements of $F_N$. We introduce the notion of a ''$(M,lambda, conjugacy class $[w]$ in $F_N$, where $Mge 1, lambda>1$ and $0<epsilon<1$. Roughly, being $(M,lambda, epsilon)$-minimal means that every $phiin Out(F_N)$ either increases the length $||w||_A$ by a factor of at least $lambda$, or distorts the length $||w||_A$ multiplicatively by a factor $epsilon$-close to $1$, and that the number of automorphically minimal $[u]$ in the orbit $Out(F_N)[w]$ is bounded by $M$. We then show that if a conjugacy class $[w]$ in $F_N$ is sufficiently close to a projective geodesic current $[nu]in PCurr(F_N)$, then, after applying a single reducing automorphism $psi=psi(nu)in Out(F_N)$ depending on $nu$ only, the element $psi([w])$ is $(M,lambda, epsilon)$-minimal for some uniform constants $M,lambda,epsilon$. Consequently, for such $[w]$, Whitehead's algorithm for the automorphic equivalence problem in $F_N$ works in quadratic time on the input $([w], [w'])$ where $[w']$ is arbitrary, and in linear time if $[w']$ is also projectively close to $[nu]$. We then show that a wide class of processes produce random conjugacy classes $[w_n]$ that projectively converge to some filling current in $PCurr(F_N)$. For such $[w_n]$ Whitehead's algorithm has at most quadratic generic-case complexity." @default.
- W2922456536 created "2019-03-22" @default.
- W2922456536 creator A5023281472 @default.
- W2922456536 date "2019-03-17" @default.
- W2922456536 modified "2023-09-27" @default.
- W2922456536 title "Generic-case complexity of Whitehead's algorithm, revisited" @default.
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