arXiv subjects

Daniel Allcock

Publications and source records attributed to Daniel Allcock.

At least 19 recordsLinked to original sources

arXiv2026

Arithmetic Polyhedra

The Koebe-Andreev-Thurston theorem assigns a 3-dimensional hyperbolic reflection group to each combinatorial polyhedron. A natural question is: which of them are arithmetic? In 2016, Kontorovich-Nakamura conjectured that all arithmetic reflection groups obtained in this way are commensurable to those obtained from the tetrahedron, square pyramid, or cuboctahedron. In this paper, we prove the conjecture. It is a consequence of the following result of independent interest: all arithmetic ideal, right-angled hyperbolic polyhedra are obtained by gluing together copies of one of three ``seed'' polyhedra.

math.MG↗

arXiv2023

The Deligne-Mostow 9-ball, and the monster

The "monstrous proposal" of the first author is that the quotient of a certain 13-dimensional complex hyperbolic braid group, by the relations that its natural generators have order 2, is the bimonster" (M x M)semidirect Z/2. Here M is the monster simple group. We prove that this quotient is either the bimonster or Z/2. In the process, we give new information about the isomorphism found by Deligne-Mostow, between the moduli space of 12-tuples in CP1 and a quotient of the complex 9-ball. Namely, we identify which loops in the 9-ball quotient correspond to the standard braid generators.

math.GT↗

arXiv2021

An Alternative to Vinberg's Algorithm

Vinberg's algorithm is the main method for finding fundamental domains for reflection groups acting on hyperbolic space. Experience shows that it can be slow. We explain why this should be expected, and prove this slowness in some cases. And we provide an alternative algorithm that should be much faster. It depends on an algorithm for finding vectors with small positive norm in indefinite binary quadratic forms, of independent interest.

math.GR↗

arXiv2021

Variations on Glauberman's ZJ Theorem

We give a new proof of Glauberman's ZJ Theorem, in a form that clarifies the choices involved and offers more choices than classical treatments. In particular, we introduce two new ZJ-type subgroups of a $p$-group $S$, that contain $ZJ_r(S)$ and $ZJ_o(S)$ respectively and are often strictly larger.

math.GR↗

arXiv2020

Most big mapping class groups fail the Tits Alternative

Let $X$ be a surface, possibly with boundary. Suppose it has infinite genus or infinitely many punctures, or a closed subset which is a disk with a Cantor set removed from its interior. For example, $X$ could be any surface of infinite type with only finitely many boundary components. We prove that the mapping class group of $X$ does not satisfy the Tits Alternative. That is, Map$(X)$ contains a finitely generated subgroup that is not virtually solvable and contains no nonabelian free group.

math.GR↗

arXiv2020

The Conway-Sloane calculus for 2-adic lattices

We motivate and explain the system introduced by Conway and Sloane for working with quadratic forms over the 2-adic integers, and prove its validity. Their system is far better for actual calculations than earlier methods, and has been used for many years, but no proof has been published before now.

math.NT↗

arXiv2019

Tropical moduli spaces as symmetric Delta-complexes

We develop techniques for studying fundamental groups and integral singular homology of symmetric Delta-complexes, and apply these techniques to study moduli spaces of stable tropical curves of unit volume, with and without marked points. As one application, we show that Delta_g and Delta_{g,n} are simply connected, for positive g. We also show that Delta_3 is homotopy equivalent to the 5-sphere, and that Delta_4 has 3-torsion in H_5.

math.AG↗

arXiv2019

Best play in Dots and Boxes endgames

We give very simple algorithms for best play in the simplest kind of Dots & Boxes endgames: those that consist entirely of loops and long chains. In every such endgame we compute the margin of victory, assuming both players maximize the number of boxes they capture, and specify a move that leads to that result. We improve on results of Buzzard and Ciere on the same problem: our algorithms examine only the current position and do not need to consider the game tree at all.

math.CO↗

arXiv2018

The tetrahedron and automorphisms of Enriques and Coble surfaces of Hessian type

Consider a cubic surface satisfying the mild condition that it may be described in Sylvester's pentahedral form. There is a well-known Enriques or Coble surface S with K3 cover birationally isomorphic to the Hessian surface of this cubic surface. We describe the nef cone and the (-2)-curves of S. In the case of pentahedral parameters (1, 1, 1, 1, nonzero t) we compute the automorphism group of S. For t not 1 it is the semidirect product of the free product (Z/2)*(Z/2)*(Z/2)*(Z/2) by the symmetric group S4. In the special case t=1/16 we study the action of Aut(S) on an invariant smooth rational curve C on the Coble surface S. We describe the action and its image, both geometrically and arithmetically. In particular, we prove that Aut(S)-->Aut(C) is injective in characteristic 0 and we identify its image with the subgroup of PGL2 coming from the symmetries of a regular tetrahedron and the reflections across its facets.

math.AG↗

arXiv2018

Congruence subgroups and Enriques surface automorphisms

We give conceptual proofs of some results on the automorphism group of an Enriques surface X, for which only computational proofs have been available. Namely, there is an obvious upper bound on the image of Aut(X) in the isometry group of X's numerical lattice, and we establish a lower bound for the image that is quite close to this upper bound. These results apply over any algebraically closed field, provided that X lacks nodal curves, or that all its nodal curves are (numerically) congruent to each other mod 2. In this generality these results were originally proven by Looijenga and Cossec-Dolgachev, developing earlier work of Coble.

math.AG↗

arXiv2017

Prenilpotent pairs in the E10 root system

Tits has defined Kac-Moody groups for all root systems, over all commutative rings with unit. A central concept is the idea of a prenilpotent pair of (real) roots. In particular, writing down his group presentation explicitly would require knowing all the Weyl-group orbits of such pairs. We show that for the hyperbolic root system E10 there are so many orbits that any attempt at direct enumeration is impractical. Namely, the number of orbits of prenilpotent pairs having inner product k grows at least as fast as (constant)(7th power of k). Our purpose is to motivate alternate approaches to Tits' groups.

math.GR↗

arXiv2017

Generators for a complex hyperbolic braid group

We give generators for a certain complex hyperbolic braid group. That is, we remove a hyperplane arrangement from complex hyperbolic $13$-space, take the quotient of the remaining space by a discrete group, and find generators for the orbifold fundamental group of the quotient. These generators have the most natural form: loops corresponding to the hyperplanes which come nearest the basepoint. Our results support the conjecture that motivated this study, the "monstrous proposal", which posits a relationship between this braid group and the monster finite simple group.

math.GT↗

arXiv2016

The densest lattices in PGL3(Q2)

We find the smallest possible covolume for lattices in PGL3(Q2), show that there are exactly two lattices with this covolume, and describe them explicitly. They are commensurable, and one of them appeared in Mumford's construction of his fake projective plane. We also discuss a new 2-adic uniformization of another fake projective plane.

math.GR↗

arXiv2016

Spherical space forms revisited

We give a simplified proof of J. A. Wolf's classification of finite groups that can act freely and isometrically on a round sphere of some dimension. We slightly improve the classification by removing some non-obvious redundancy. The groups are the same as the Frobenius complements of finite group theory.

math.GT↗

arXiv2016

Steinberg groups as amalgams

For any root system and any commutative ring we give a relatively simple presentation of a group related to its Steinberg group St. This includes the case of infinite root systems used in Kac-Moody theory, for which the Steinberg group was defined by Tits and Morita-Rehmann. In most cases our group equals St, giving a presentation with many advantages over the usual presentation of St. This equality holds for all spherical root systems, all irreducible affine root systems of rank>2, and all 3-spherical root systems. When the coefficient ring satisfies a minor condition, the last condition can be relaxed to 2-sphericity. Our presentation is defined in terms of the Dynkin diagram rather than the full root system. It is concrete, with no implicit coefficients or signs. It makes manifest the exceptional diagram automorphisms in characteristics 2 and 3, and their generalizations to Kac-Moody groups. And it is a Curtis-Tits style presentation: it is the direct limit of the groups coming from 1- and 2-node subdiagrams of the Dynkin diagram. Over non-fields this description as a direct limit is new and surprising. Our main application is that many Steinberg and Kac-Moody groups over finitely-generated rings are finitely presented.

math.GR↗

arXiv2015

Presentation of hyperbolic Kac-Moody groups over rings

Tits has defined Kac-Moody and Steinberg groups over commutative rings, providing infinite dimensional analogues of the Chevalley-Demazure group schemes. Here we establish simple explicit presentations for all Steinberg and Kac-Moody groups whose Dynkin diagrams are hyperbolic and simply laced. Our presentations are analogues of the Curtis-Tits presentation of the finite groups of Lie type. When the ground ring is finitely generated, we derive the finite presentability of the Steinberg group, and similarly for the Kac-Moody group when the ground ring is a Dedekind domain of arithmetic type. These finite-presentation results need slightly stronger hypotheses when the rank is smallest possible, namely 4. The presentations simplify considerably when the ground ring is Z, a case of special interest because of the conjectured role of the Kac-Moody group E10(Z) in superstring theory.

math.GR↗

arXiv2015

Presentation of affine Kac-Moody groups over rings

Tits has defined Steinberg groups and Kac-Moody groups for any root system and any commutative ring R. We establish a Curtis-Tits-style presentation for the Steinberg group St of any rank > 2 irreducible affine root system, for any R. Namely, St is the direct limit of the Steinberg groups coming from the 1- and 2-node subdiagrams of the Dynkin diagram. This leads to a completely explicit presentation. Using this we show that St is finitely presented if the rank is > 3 and R is finitely generated as a ring, or if the rank is 3 and R is finitely generated as a module over a subring generated by finitely many units. Similar results hold for the corresponding Kac-Moody groups when R is a Dedekind domain of arithmetic type.

math.GR↗

arXiv2015

Geometric generators for braid-like groups

We study the problem of finding generators for the fundamental group G of a space of the following sort: one removes a family of complex hyperplanes from n dimensional complex vector space, or n dimensional complex hyperbolic space, or the Hermitian symmetric space for O(2,n), and then takes the quotient by a discrete group $PΓ$. The classical example is the braid group, but there are many similar "braid-like" groups that arise in topology and algebraic geometry. Our main result is that if $PΓ$ contains reflections in the hyperplanes nearest the basepoint, and these reflections satisfy a certain property, then G is generated by the analogues of the generators of the classical braid group. We apply this to obtain generators for G in a particular intricate example in complex hyperbolic space of dimension 13. The interest in this example comes from a conjectured relationship between this braid-like group and the monster simple group M, that gives geometric meaning to the generators and relations in the Conway-Simons presentation of $(M \times M):2$.

math.GT↗