Let $X$ be an orbifold that is a global quotient of a manifold $Y$ by a finite group $G$. We construct a noncommutative ring $H\sp \ast(Y, G)$ with a $G$-action such that $H\sp*(Y, G)\sp G$ is the orbifold cohomology ring of $X$ defined by W. Chen and Y. Ruan [CR]. When $Y=S\sp n$, with $S$ a surface with trivial canonical class and $G = \mathfrak {S}\sb n$, we prove that (a small modification of) the orbifold cohomology of $X$ is naturally isomorphic to the cohomology ring of the Hilbert scheme $S\sp {[n]}$, computed by M. Lehn and C. Sorger [LS2].

We show that the sectional capacity of an adelic subset of a projective variety over a number field is a quasi-canonical limit of arithmetic top self-intersection numbers, and we establish the functorial properties of extremal plurisubharmonic Green's functions. We also present a conjecture that the sectional capacity should be a top selfintersection number in an appropriate adelic arithmetic intersection theory.

Let $X$ be a scheme; the fundamental group scheme of $X$, when it exists, is a profinite group scheme that classifies principal homogeneous spaces under finite flat group schemes over $X$. We generalize the construction of the fundamental group scheme given by M. Nori [No] to the case when $X$ is a reduced flat scheme over a Dedekind scheme. We prove that if $X$ is a curve over a $p$-adic field having good reduction, then the prime-to-$p$ part of the fundamental group scheme of $X$ has only finitely many rational representations in ${\rm GL}\sb N$. In the second part of the paper, using tools from Arakelov theory, we construct an intrinsic height on the moduli space of semistable vector bundles (of fixed rank and degree) over a curve defined over a number field. We finally prove that the height of vector bundles over an arithmetic surface $X$ coming from representations of the fundamental group scheme is upper bounded; so we deduce that there are only finitely many isomorphism classes of rational representations of the fundamental group scheme of $X$.

Let $ 1< p < \infty$. It is shown that if $G$ is a discrete group with the approximation property introduced by U. Haagerup and J. Kraus, then the noncommutative $L\sb p(VN(G))$-space has the operator space approximation property. If, in addition, the group von Neumann algebra $VN(G)$ has the quotient weak expectation property (QWEP), that is, is a quotient of a $C\sp \ast$-algebra with Lance's weak expectation property, then $L\sb p(V N(G))$ actually has the completely contractive approximation property and the approximation maps can be chosen to be finite-rank completely contractive multipliers on $L\sb p(V N(G))$. Finally, we show that if $G$ is a countable discrete group having the approximation property and $V N(G)$ has the QWEP, then $L\sb p(V N(G))$ has a very nice local structure; that is, it is a $\mathscr {COL}\sb p$-space and has a completely bounded Schauder basis.

There is a natural conjecture that the universal bounds for the dimension spectrum of harmonic measure are the same for simply connected and for nonsimply connected domains in the plane. Because of the close relation to conformal mapping theory, the simply connected case is much better understood, and proving the above statement would give new results concerning the properties of harmonic measure in the general case.

We establish the conjecture in the category of domains bounded by polynomial Julia sets. The idea is to consider the coefficients of the dynamical zeta function as subharmonic functions on a slice of Teichmüller's space of the polynomial and then to apply the maximum principle.

We prove a new uniform bound for subgroup growth of a Chevalley group $G$ over the local ring $\mathbb {F}[[t]]$ and also over local pro-$p$ rings of higher Krull dimension. This is applied to the determination of congruence subgroup growth of arithmetic groups over global fields of positive characteristic. In particular, we show that the subgroup growth of ${\rm SL}\sb n(F\sb p[t]) (n\geq3)$ is of type $n\sp {\log n}$. This was one of the main problems left open by A. Lubotzky in his article [5].

The essential tool for proving the results is the use of graded Lie algebras. We sharpen Lubotzky's bounds on subgroup growth via a result on subspaces of a Chevalley Lie algebra $L$ over a finite field $\mathbb {F}$. This theorem is proved by algebraic geometry and can be modified to obtain a lower bound on the codimension of proper Lie subalgebras of $L$.

After correcting a statement in [2], we indicate how to obtain a more precise result concerning fundamental solutions to the Tricomi operator.