2 research outputs found
Listing Words in Free Groups
Lists of equivalence classes of words under rotation or rotation plus
reversal (i.e., necklaces and bracelets) have many uses, and efficient
algorithms for generating these lists exist. In combinatorial group theory
elements of a group are typically written as words in the generators and their
inverses, and necklaces and bracelets correspond to conjugacy classes and
relators respectively. We present algorithms to generate lists of freely and
cyclically reduced necklaces and bracelets in free groups. Experimental
evidence suggests that these algorithms are CAT -- that is, they run in
constant amortized time
Walks on Free Groups and other Stories -- twelve years later
We start by studying the distribution of (cyclically reduced) elements of the
free groups Fn with respect to their abelianization (or equivalently, their
integer homology class. We derive an explicit generating function, and a
limiting distribution, by means of certain results (of independent interest) on
Chebyshev polynomials; we also prove that the reductions modulo an arbitrary
prime of these classes are asymptotically equidistributed, and we study the
deviation from equidistribution. We extend our techniques to a more general
setting and use them to study the statistical properties of long cycles (and
paths) on regular (directed and undirected) graphs. We return to the free group
to study some growth functions of the number of conjugacy classes as a function
of their cyclically reduced length.Comment: 45pp, appeared in the Schupp volume of the Illinois Journal of
Mathematics, published version of arXiv:math/991107