Manin’s conjecture predicts the rate of growth of rational points of a bounded height after removing those lying on an exceptional set. We study whether the exceptional set in Manin’s conjecture is a thin set using the minimal model program and boundedness of log Fano varieties.

We define and study the stack ${\mathcal{U}}_{g,g}^{ns,a}$ of (possibly singular) projective curves of arithmetic genus $g$ with $g$ smooth marked points forming an ample nonspecial divisor. We define an explicit closed embedding of a natural ${\mathbb{G}}_m^g$-torsor ${\tilde{\mathcal{U}}}_{g,g}^{ns,a}$ over ${\mathcal{U}}_{g,g}^{ns,a}$ into an affine space, and we give explicit equations of the universal curve (away from characteristics $2$ and $3$). This construction can be viewed as a generalization of the Weierstrass cubic and the $j$-invariant of an elliptic curve to the case $g>1$. Our main result is that in characteristics different from $2$ and $3$ the moduli space ${\tilde{\mathcal{U}}}_{g,g}^{ns,a}$ is isomorphic to the moduli space of minimal $A_{\infty }$-structures on a certain finite-dimensional graded associative algebra $E_g$ (introduced by Fisette and Polishchuk).

We investigate the rank gradient and growth of torsion in homology in residually finite groups. As a tool, we introduce a new complexity notion for generating sets, using measured groupoids and combinatorial cost. As an application we prove the vanishing of the above invariants for Farber sequences of subgroups of right-angled groups. A group is right angled if it can be generated by a sequence of elements of infinite order such that any two consecutive elements commute.

Most nonuniform lattices in higher rank simple Lie groups are right angled. We provide the first examples of uniform (cocompact) right-angled arithmetic groups in $\mathrm{SL}(n,\mathbb{R})$, $n\ge 3$, and $\mathrm{SO}(p,q)$ for some values of $p,q$. This is a class of lattices for which the congruence subgroup property is not known in general. By using rigidity theory and the notion of invariant random subgroups it follows that both the rank gradient and the homology torsion growth vanish for an arbitrary sequence of subgroups in any right-angled lattice in a higher rank simple Lie group.

We introduce a new product for permutation groups. It takes as input two permutation groups, $M$ and $N$ and produces an infinite group $M⊠N$ which carries many of the permutational properties of $M$. Under mild conditions on $M$ and $N$ the group $M⊠N$ is simple.

As a permutational product, its most significant property is the following: $M⊠N$ is primitive if and only if $M$ is primitive but not regular, and $N$ is transitive. Despite this remarkable similarity with the wreath product in product action, $M⊠N$ and $MWrN$ are thoroughly dissimilar.

The product provides a general way to build exotic examples of nondiscrete, simple, totally disconnected, locally compact, compactly generated topological groups from discrete groups.

We use this to obtain the first construction of uncountably many pairwise nonisomorphic simple topological groups that are totally disconnected, locally compact, compactly generated, and nondiscrete. The groups we construct all contain the same compact open subgroup. The analogous result for discrete groups was proved in 1953 by Ruth Camm.

To build the product, we describe a group $\mathcal{U}(M,N)$ that acts on a biregular tree $T$. This group has a natural universal property and is a generalization of the iconic universal group construction of Marc Burger and Shahar Mozes for locally finite regular trees.