Teaching Statement and Statement of Research Plans - GWDG

4 Index
(with respect to a certain set of generators), is the same as the Cayley graph
of a well-known group PGL(d) over a finite field, with respect to a set of
easily-described generators.
Property (τ), Selberg Property, Ramanujan-Peterson conjecture
It is well known that every arithmetic lattice Γ (at least over fields of characteristic
0 and conjecturally always) has the Selberg property. If in addition
Γ has the congruence subgroup property, then Γ has also (τ). This leads to
following conjecture: see [16] for more detail.
Conjecture 1. Let Γ be a lattice in a semisiple group.
congruence subgroup property if and only if Γ has (τ).
Then Γ has the
There is a close relationship between the congruence subgroup property, the
Selberg property and the Ramanujan-Peterson conjecture. One of my main
goals is to study these relationships more deeply with the viewpoint of obtaining
more information related to the above conjecture. I have some ideas
how to generalise the known case to the higher dimensional case and I am
interested in arithmetic groups having property (τ) over a division ring and
I want to consider the effect of the (τ) property on the lifting of the representations
to the similar corresponding linear groups such as GL or SL under
the Langlands corespondence.
Group action on the set of galleries of a Bruhat-Tits Building
Let D be a division algebra over a function field K i.e, a finite extension
of F q (t) with dim D K = d 2 , and π = (π) a place which is unramified in D,
i.e,
D π := D ⊗ K K π
∼ = Md (K π )
and π is an irreducible polynomial in OsupsetF q [t] .
Let M 0 and M π 0 be the set of maximal orders in D and D π respectively. We
can see first that,M π 0 can be interprated as the set of vertices of a Bruhat-
Tits building X π ∗ associated to PGL d (K π ). We can also consider another
Bruhat-Tits building X (global case) with M 0 as the set of vertices. Let R
be a fix maximal order in D. Now consider the set:
The unit group Γ (1)
R
X 0 (R) π = {M ∈ M 0 | M v = R v for all v ≠ π}
R is equal to the number of orbits of Γ (1)
R
:= R ⊗ O[ 1
π ]× acts on X 0 (R) π and the class number of
under this action. For any oder S