We prove that if a closed hyperbolic $3$ -manifold M contains infinitely many totally geodesic surfaces, then M is arithmetic.

We prove that the sign of the Euler characteristic of arithmetic groups with the congruence subgroup property is determined by the profinite completion. In contrast, we construct examples showing that this is not true for the Euler characteristic itself and that the sign of the Euler characteristic is not profinite among general residually finite groups of type F. Our methods imply similar results for $\ell^2$ -torsion as well as a strong profiniteness statement for Novikov–Shubin invariants.

Let G be a semisimple real algebraic group defined over ${\mathbb {Q}}$ , $\Gamma $ be an arithmetic subgroup of G, and T be a maximal ${\mathbb {R}}$ -split torus. A trajectory in $G/\Gamma $ is divergent if eventually it leaves every compact subset. In some cases there is a finite collection of explicit algebraic data which accounts for the divergence. If this is the case, the divergent trajectory is called obvious. Given a closed cone in T, we study the existence of non-obvious divergent trajectories under its action in $G\kern-1pt{/}\kern-1pt\Gamma $ . We get a sufficient condition for the existence of a non-obvious divergence trajectory in the general case, and a full classification under the assumption that $\mathrm {rank}_{{\mathbb {Q}}}G=\mathrm {rank}_{{\mathbb {R}}}G=2$ .

A question of Griffiths–Schmid asks when the monodromy group of an algebraic family of complex varieties is arithmetic. We resolve this in the affirmative for a class of algebraic surfaces known as Atiyah–Kodaira manifolds, which have base and fibers equal to complete algebraic curves. Our methods are topological in nature and involve an analysis of the ‘geometric’ monodromy, valued in the mapping class group of the fiber.

We prove two results about the width of words in $\operatorname{SL}_{n}(\mathbb{Z})$. The first is that, for every $n\geqslant 3$, there is a constant $C(n)$ such that the width of any word in $\operatorname{SL}_{n}(\mathbb{Z})$ is less than $C(n)$. The second result is that, for any word $w$, if $n$ is big enough, the width of $w$ in $\operatorname{SL}_{n}(\mathbb{Z})$ is at most 87.

We determine the index of five of the seven hypergeometric Calabi–Yau operators that have finite index in $\mathit{Sp}_{4}(\mathbb{Z})$ and in two cases give a complete description of the monodromy group. Furthermore, we find six nonhypergeometric Calabi–Yau operators with finite index in $\mathit{Sp}_{4}(\mathbb{Z})$, most notably a case where the index is one.

We apply the endoscopic classification of automorphic forms on $U(3)$ to study the growth of the first Betti number of congruence covers of a Picard modular surface. As a consequence, we establish a case of a conjecture of Sarnak and Xue on cohomology growth.

The work of Reid, Chinburg–Hamilton–Long–Reid, Prasad–Rapinchuk, and the author with Reid have demonstrated that geodesics or totally geodesic submanifolds can sometimes be used to determine the commensurability class of an arithmetic manifold. The main results of this article show that generalizations of these results to other arithmetic manifolds will require a wide range of data. Specifically, we prove that certain incommensurable arithmetic manifolds arising from the semisimple Lie groups of the form ${{\left( \text{SL(d,}\,\mathbf{R}) \right)}^{r}}\,\times \,{{\left( \text{SL(d,}\,\mathbf{C}) \right)}^{s}}$ have the same commensurability classes of totally geodesic submanifolds coming from a fixed field. This construction is algebraic and shows the failure of determining, in general, a central simple algebra from subalgebras over a fixed field. This, in turn, can be viewed in terms of forms of $\text{S}{{\text{L}}_{d}}$ and the failure of determining the form via certain classes of algebraic subgroups.

The normal residual finiteness growth of a group quantifies how well approximated the group is by its finite quotients. We show that any S-arithmetic subgroup of a higher rank Chevalley group G has normal residual finiteness growth ndim (G).

Consider the group scheme

$$ R \mapsto G(R) := R \rtimes R^* = \left( \begin{array}{cc} 1 & R \\ 0 & R^* \end{array} \right) = \left( \begin{array}{cc} 1 & *\\ 0 & * \end{array} \right) \cap \mathrm{GL}_2(R) $$

where $R$ is an arbitrary commutative ring with $1 \neq 0$ and a unit $x \in R^*$ acts on $R$ by multiplication.

We will study the finiteness properties of subgroups of $G(\Oka_S)$ where $\Oka_S$ is an $S$-arithmetic subring of a global function field. The subgroups we are interested in are of the form $\Oka_S \rtimes Q$ where $Q$ is a subgroup of $\Oka_S^*$. The finiteness properties of these metabelian groups can be expressed in terms of the $\Sigma$-invariant due to R. Bieri and R. Strebel.

{\sc Theorem A.} {\it Let $S$ be a finite set of places of a global function field (regarded as normalized discrete valuations) and $\Oka_S$ the corresponding $S$-arithmetic ring. Let $Q$ be a subgroup of $\Oka_S^*$. Then $Q$ is finitely generated and for all integers $n \geq 1$ the following are equivalent:

\begin{enumerate} \item[\rm (1)] $\Oka_S \rtimes Q$ is of type FP$_n$;

\item[\rm (2)] $\Oka_S$ is $n$-tame as a $\ZZZ Q$-module;

\item[\rm (3)] each $p \in S$ restricts to a non-trivial homomorphism $p|_{Q} : Q \rightarrow \RRR$ and the set $\{ [p|_Q] \mid p \in S \} \subseteq \SSS(Q)$ is $n$-tame. \end{enumerate}

If these conditions hold for at least one $n \geq 1$ then the identity

$$ \Sigma_{\Oka_S}^c(Q,\ZZZ) = \{ [p|_Q] \mid p \in S \} $$

holds.} {\sc Theorem B.} {\it Let $r$ denote the rank of $Q$. Then the following hold: \begin{enumerate} \item[\rm (1)] the group $\Oka_S \rtimes Q$ is not of type FP$_{r+1}$;

\item[\rm (2)] if $Q$ has maximum rank $r=|S|-1$ then the group $\Oka_S \rtimes Q$ is of type FP$_r$.

\end{enumerate} In particular, $G(\Oka_S) = \Oka_S \rtimes \Oka_S^*$ is of type FP$_{|S|-1}$ but not of type FP$_{|S|}$.}

1991 Mathematics Subject Classification: 20E08, 20F16, 20G30, 52A20.