34 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
Ramanujan Coverings of Graphs
Let be a finite connected graph, and let be the spectral radius of
its universal cover. For example, if is -regular then
. We show that for every , there is an -covering
(a.k.a. an -lift) of where all the new eigenvalues are bounded from
above by . It follows that a bipartite Ramanujan graph has a Ramanujan
-covering for every . This generalizes the case due to Marcus,
Spielman and Srivastava (2013).
Every -covering of corresponds to a labeling of the edges of by
elements of the symmetric group . We generalize this notion to labeling
the edges by elements of various groups and present a broader scenario where
Ramanujan coverings are guaranteed to exist.
In particular, this shows the existence of richer families of bipartite
Ramanujan graphs than was known before. Inspired by Marcus-Spielman-Srivastava,
a crucial component of our proof is the existence of interlacing families of
polynomials for complex reflection groups. The core argument of this component
is taken from a recent paper of them (2015).
Another important ingredient of our proof is a new generalization of the
matching polynomial of a graph. We define the -th matching polynomial of
to be the average matching polynomial of all -coverings of . We show this
polynomial shares many properties with the original matching polynomial. For
example, it is real rooted with all its roots inside .Comment: 38 pages, 4 figures, journal version (minor changes from previous
arXiv version). Shortened version appeared in STOC 201