Papers by Donald Cartwright
arXiv (Cornell University), Jan 10, 2017
In order to enumerate the fake projective planes, as announced in [2], we found explicit generato... more In order to enumerate the fake projective planes, as announced in [2], we found explicit generators and a presentation for each maximal arithmetic subgroupΓ of P U (2, 1) for which the (appropriately normalized) covolume equals 1/N for some integer N ≥ 1. Prasad and Yeung [4, 5] had given a list of all suchΓ (up to equivalence). The generators were found by a computer search which uses the natural action of P U (2, 1) on the unit ball B(C 2) in C 2. Our main results here give criteria which ensure that the computer search has found sufficiently many elements ofΓ to generateΓ, and describes a family of relations amongst the generating set sufficient to give a presentation ofΓ. We give an example illustrating details of how this was done in the case of a particularΓ (for which N = 864). While there are no fake projective planes in this case, we exhibit a torsion-free subgroup Π of index N inΓ, and give some properties of the surface Π\B(C 2).
arXiv (Cornell University), Sep 26, 2005
A network is a countable, connected graph X viewed as a one-complex, where each edge [x, y] = [y,... more A network is a countable, connected graph X viewed as a one-complex, where each edge [x, y] = [y, x] (x, y ∈ X 0 , the vertex set) is a copy of the unit interval within the graph's one-skeleton X 1 and is assigned a positive conductance c(xy). A reference "Lebesgue" measure on X 1 is built up by using Lebesgue measure with total mass c(xy) on each edge [x, y]. There are three natural operators on X: the transition operator P acting on functions on X 0 (the reversible Markov chain associated with c), the averaging operator A over spheres of radius 1 on X 1 , and the Laplace operator ∆ on X 1 (with Kirchhoff conditions weighted by c at the vertices). The relation between the ℓ 2-spectrum of P and the H 2-spectrum of ∆ was described by Cattaneo [4]. In this paper we describe the relation between the ℓ 2-spectrum of P and the L 2-spectrum of A.
Groups acting simply transitively on the vertices of a building of type à n
Cambridge University Press eBooks, Jan 12, 1995
arXiv (Cornell University), Apr 21, 1994
We see that a building whose Coxeter group is hyperbolic is itself hyperbolic. Thus any finitely ... more We see that a building whose Coxeter group is hyperbolic is itself hyperbolic. Thus any finitely generated group acting co-compactly on such a building is hyperbolic, hence automatic. We turn our attention to affine buildings and consider a group Γ which acts simply transitively and in a "type-rotating" way on the vertices of a locally finite thick building of type A n. We show that Γ is biautomatic, using a presentation of Γ and unique normal form for each element of Γ, as described in [5].
arXiv (Cornell University), Dec 12, 2014
In this article, we study various concrete algebraic and differential geometric properties of the... more In this article, we study various concrete algebraic and differential geometric properties of the Cartwright-Steger surface, the unique smooth surface of Euler number 3 which is neither a projective plane nor a fake projective plane. In particular, we determine the genus of a generic fiber of the Albanese fibration, and deduce that the singular fibers are not totally geodesic, answering an open problem about fibrations of a complex ball quotient over a Riemann surface.
A ne Buildings and Automatic Groups
A family of � n -groups
Isr J Math, 1998
GROUPS A CTING S IMPLY T RANSITIVELY O N T HE VERTICES O F A BUILDING O F T YPE Az, II: THE C ASES q =2 A ND q =3
Geom Dedic, 1993
Singularities of the Green''s function of a random walk on a discrete group
Harmonic functions on buildings of A ˜ n
L//p-norms of characters on the exceptional compact Lie groups
Bollettino della Unione Matematica Italiana B
On the Cartwright-Steger surface
In this article, we study various concrete algebraic and differential geometric properties of the... more In this article, we study various concrete algebraic and differential geometric properties of the Cartwright-Steger surface. In particular, we determine the genus of a generic fiber of the Albanese fibration, and deduce that the singular fibers are not totally geodesic, answering an open problem about fibrations of a complex ball quotient over a Riemann surface
Construction of lattices defining fake projective planes - Lecture 1
A network is a countable, connected graph X viewed as a one-complex, where each edge [x, y] = [y,... more A network is a countable, connected graph X viewed as a one-complex, where each edge [x, y] = [y, x] (x, y ∈ X 0 , the vertex set) is a copy of the unit interval within the graph's one-skeleton X 1 and is assigned a positive conductance c(xy). A reference "Lebesgue" measure on X 1 is built up by using Lebesgue measure with total mass c(xy) on each edge [x, y]. There are three natural operators on X: the transition operator P acting on functions on X 0 (the reversible Markov chain associated with c), the averaging operator A over spheres of radius 1 on X 1 , and the Laplace operator ∆ on X 1 (with Kirchhoff conditions weighted by c at the vertices). The relation between the ℓ 2-spectrum of P and the H 2-spectrum of ∆ was described by Cattaneo [4]. In this paper we describe the relation between the ℓ 2-spectrum of P and the L 2-spectrum of A.
Illinois Journal of Mathematics, 2007
A network is a countable, connected graph X viewed as a one-complex, where each edge [x, y] = [y,... more A network is a countable, connected graph X viewed as a one-complex, where each edge [x, y] = [y, x] (x, y ∈ X 0 , the vertex set) is a copy of the unit interval within the graph's one-skeleton X 1 and is assigned a positive conductance c(xy). A reference "Lebesgue" measure on X 1 is built up by using Lebesgue measure with total mass c(xy) on each edge [x, y]. There are three natural operators on X: the transition operator P acting on functions on X 0 (the reversible Markov chain associated with c), the averaging operator A over spheres of radius 1 on X 1 , and the Laplace operator ∆ on X 1 (with Kirchhoff conditions weighted by c at the vertices). The relation between the ℓ 2-spectrum of P and the H 2-spectrum of ∆ was described by Cattaneo [4]. In this paper we describe the relation between the ℓ 2-spectrum of P and the L 2-spectrum of A.
Nagoya Mathematical Journal, 1986
Suppose that G is a discrete group and p is a probability measure on G. Consider the associated r... more Suppose that G is a discrete group and p is a probability measure on G. Consider the associated random walk {Xn} on G. That is, let Xn = Y1Y2 … Yn, where the Yj’s are independent and identically distributed G-valued variables with density p. An important problem in the study of this random walk is the evaluation of the resolvent (or Green’s function) R(z, x) of p. For example, the resolvent provides, in principle, the values of the n step transition probabilities of the process, and in several cases knowledge of R(z, x) permits a description of the asymptotic behaviour of these probabilities.
Enumeration of the 50 fake projective planes = Énumération des 50 faux plans projectifs
Building upon the classification of Prasad and Yeung [Invent. Math. 168 (2007) 321–370], we have ... more Building upon the classification of Prasad and Yeung [Invent. Math. 168 (2007) 321–370], we have shown that there exist exactly 50 fake projective planes (up to homeomorphism; 100 up to biholomorphism), and exhibited each of them explicitly as a quotient of the unit ball in C 2 . Some of these fake planes admit singular quotients by 3 element groups and three of these quotients are simply connected. Also exhibited are algebraic surfaces with c 2 1 =3 c 2=9 n for any positive integer n . En partant de la classification de Prasad et Yeung [Invent. Math. 168 (2007) 321–370], nous montrons qu'il existe precisement 50 faux plans projectifs (a homeomorphisme pres, 100 a biholomorphisme pres), et presentons chacun comme un quotient de la boule unite de C 2 . Certains de ces plans admettent des quotients singuliers par des groupes d'automorphismes a 3 elements, et trois d'entre eux sont simplement connexes. De plus, pour chaque entier n >0, nous presentons des surfaces algebriques avec c 2 1 =3 c 2=9 n .
Ramanujan geometries of type �n
Dm, 2003
Tensor product of spherical representations of the group of automorphisms of a homogeneous tree
Correlations and Characterizations of the Uniform Distribution
Australian Journal of Statistics, 1986
ABSTRACT Two characterizations of the uniform distribution on a suitable compact space are proved... more ABSTRACT Two characterizations of the uniform distribution on a suitable compact space are proved. These characterizations are applied to a number of particular examples of which the most interesting is the following: if X, Y and Z are independent n-vectors whose components are independent and identically distributed within a vector, then the pairwise independence of the product moment correlation coefficients between X, Y and Z implies that these vectors are normally distributed.
Uploads
Papers by Donald Cartwright