Nikolai Vavilov, St. Petersburg
“The Yoga of Commutators (joint work with Roozbeh Hazrat, Alexei Stepanov and Zuhong Zhang)”
In
an
abstract
group,
an
element
of
the
commutator
subgroup
is
not
necessarily
a
commutator.
However,
the
famous
Ore
conjecture,
recently
completely
settled
by
Ellers—Gordeev
and
by
Liebeck—O’Brien—Shalev—Tiep,
asserts
that
any
element
of
a
finite
simple
group
is
a
single
commutator.
On
the
other
hand,
from
the
work
of
van
der
Kallen,
Dennis
and
Vaserstein
it
is
known
that
nothing
like
that
can
possibly
hold
in
general,
for
commutators
in
classical
groups
over
rings.
Actually,
these
groups
do
not
even
have
bounded
width
with
respect
to
commutators.
In
the
present
talk,
we
report
the
amazing
recent
results
which
assert
that
exactly
the
opposite
holds:
over
any
commutative
ring
commutators
have
bounded
width
with
respect
to
elementary
generators,
which
in
the
case
of
SLn
are
the
usual
elementary
transformations
of
the
undergraduate
linear
algebra
course.
Technically,
these
results
are
based
on
a
further
development
of
localisation
methods
proposed
in
the
groundbreaking
work
by
Quillen
and
Suslin
to
solve
Serre’s
conjecture,
their
expantion
and
refinement
proposed
by
Bak,
localisation-completion,
further
enhancements
implemented
by
the
authors
(R.H.,
N.V,
and
Z.Zh.),
and
the
terrific
recent
method
of
universal
localisation,
devised
by
one
of
us
(A.S.)
Apart
from
the
above
results
on
bounded
width
of
commutators,
and
their
relative
versions,
these
new
methods
have
a
whole
range
of
further
applications,
nilpotency
of
K1,
multiple
commutator
formulae
and
the
like,
which
enhance
and
generalise
many
important
results
of
classical
algebraic
K-theory.
However,
the
talk
is
addressed
to
a
general
audience,
so
that
we
do
not
expect
serious
previous
exposure
to
algebraic
K-theory,
or
the
theory
of
algebraic
groups.
In
fact,
our
results
are
already
new
for
the
group
SLn,
and
we
plan
to
concentrate
on
background,
general
mathematical
significance
and
highlights
of
the
theory,
further
related
width
problems
(unipotent
factorisations,
powers,
etc.)
and
connections
with
geometry,
arithmetics,
and
asymptotic
group
theory,
rather
than
actual
technical
details
of
the
proofs.
There
are
also
recent
connections
with
complex
analysis
(the
work
of
Ivarsson—Kutzschebauch),
etc.