Taking up the terminology established in the first lecture, in
1970 Grothendieck showed that when two groups (G,H) form a
Grothendieck pair, there is an equivalence of their linear
representations. For recent work showing that certain groups
are...
If a monomorphism of abstract groups H↪G induces an isomorphism
of profinite completions, then (G,H) is called a Grothendieck pair,
recalling the fact that Grothendieck asked about the existence of
such pairs with G and H finitely presented and...
We discuss a local to global profinite principle for being a
commutator in some arithmetic groups. Specifically we show that
SL2(Z) satisfies such a principle, while it can fail with
infinitely many exceptions for SL2(Z[1/p]). The source of
the...
We discuss a local to global profinite principle for being a
commutator in some arithmetic groups. Specifically we show that
SL2(Z) satisfies such a principle, while it can fail with
infinitely many exceptions for SL2(Z[1/p]). The source of
the...
We explain our proof of the unbounded denominators conjecture.
This talk will require the main theorem of the lecture on Nov. 17,
2021, as a “black box” but otherwise be logically independent of
that talk.
Let f=∑anxn∈ℚ[x] be a power series which is also a meromorphic
function in some neighborhood of the origin. The subject of the
talk will be how certain conditions on f(x) as a meromorphic
function actually guarantee that f(x) is an algebraic...
Somehow, despite the title, SL(2,Z) is the poster child for
arithmetic groups not satisfying the congruence subgroup property,
which is to say that it has finite index subgroups which can not be
defined by congruence conditions on their coefficients...
After surveying some important consequences of the property of
bounded generation (BG) dealing with SS-rigidity, the congruence
subgroup problem, etc., we will focus on examples of boundedly
generated groups. We will prove that every unimodular (n×n...
I'll describe a method for analyzing the first-order theory of
an arithmetic group using its congruence quotients. When this
method works, it gives a strong form of first-order rigidity
together with a complete description of the collection of...
