10 research outputs found
Surface Words are Determined by Word Measures on Groups
Every word in a free group naturally induces a probability measure on
every compact group . For example, if is the commutator
word, a random element sampled by the -measure is given by the commutator
of two independent, Haar-random elements of . Back in
1896, Frobenius showed that if is a finite group and an irreducible
character, then the expected value of is
. This is true for any compact group, and
completely determines the -measure on these groups. An
analogous result holds with the commutator word replaced by any surface word.
We prove a converse to this theorem: if induces the same measure as
on every compact group, then, up to an automorphism of the
free group, is equal to . The same holds when
is replaced by any surface word.
The proof relies on the analysis of word measures on unitary groups and on
orthogonal groups, which appears in separate papers, and on new analysis of
word measures on generalized symmetric groups that we develop here.Comment: 16 pages, fixed the proof of Theorem 3.6, updated reference
Primitive Words, Free Factors and Measure Preservation
Let F_k be the free group on k generators. A word w \in F_k is called
primitive if it belongs to some basis of F_k. We investigate two criteria for
primitivity, and consider more generally, subgroups of F_k which are free
factors.
The first criterion is graph-theoretic and uses Stallings core graphs: given
subgroups of finite rank H \le J \le F_k we present a simple procedure to
determine whether H is a free factor of J. This yields, in particular, a
procedure to determine whether a given element in F_k is primitive.
Again let w \in F_k and consider the word map w:G x G x ... x G \to G (from
the direct product of k copies of G to G), where G is an arbitrary finite
group. We call w measure preserving if given uniform measure on G x G x ... x
G, w induces uniform measure on G (for every finite G). This is the second
criterion we investigate: it is not hard to see that primitivity implies
measure preservation and it was conjectured that the two properties are
equivalent. Our combinatorial approach to primitivity allows us to make
progress on this problem and in particular prove the conjecture for k=2.
It was asked whether the primitive elements of F_k form a closed set in the
profinite topology of free groups. Our results provide a positive answer for
F_2.Comment: This is a unified version of two manuscripts: "On Primitive words I:
A New Algorithm", and "On Primitive Words II: Measure Preservation". 42
pages, 14 figures. Some parts of the paper reorganized towards publication in
the Israel J. of Mat
Word Measures on Symmetric Groups
Fix a word in a free group on generators. A -random
permutation in the symmetric group is obtained by sampling
independent uniformly random permutations and evaluating . In
[arXiv:1104.3991, arXiv:1202.3269] it was shown that the average number of
fixed points in a -random permutation is
, where is
the smallest rank of a subgroup containing as a non-primitive
element. We show that plays a role in estimates of all
"natural" families of characters of symmetric groups: those corresponding to
"stable" representations. In particular, we show that for all , the
average number of -cycles is
. As an application, we prove
that for every , every and every large enough , Schreier
graphs with random generators depicting the action of on
-tuples, have second eigenvalue at most
asymptotically almost surely. An important ingredient in this work is a
systematic study of not-necessarily connected Stallings core graphs.Comment: 50 pages, 2 figures. Extended abstract accepted to FPSAC 2020. Added
new Appendix A. Improved introductio