Ehsaan Hossain, Department of Pure Mathematics, University of Waterloo
"Problems on sofic and hyperlinear groups"
A
sofic
group
is
one
that
can
be
locally
approximated
by
finite
groups;
it
is
equivalent
to
embed
into
a
"metric
ultraproduct''
of
finite
groups.
A
hyperlinear
group
is
the
same
thing
except
with
unitary
matrix
groups
instead
of
finite
ones.
So
every
sofic
group
is
hyperlinear,
but
it
is
still
standing
whether
these
classes
are
the
same.
In
fact
nobody
knows
even
one
example
of
a
group
that
is
not
sofic
or
hyperlinear!
Sofic/hyperlinear
groups
are
amongst
the
largest
classes
of
groups
for
which
various
famous
problems
have
been
verified:
Connes's
embedding
problem,
Gottschalk
surjunctivity,
and
Kaplansky's
group
rings
problems,
amongst
others
---
so
the
problems
in
soficity
can
give
solutions
for
even
some
of
the
biggest
conjectures
in
other
subjects.
I'll
learn
about
these
and
let
you
know
what
I
find,
and
as
the
semester
progresses,
let's
try
to
get
our
hands
dirty
with
these
groups
and
see
what
the
field
is
like.
MC 5479