Bernard Maskit Memorial Tribute
This is an expanded version of the Maskit memorial tribute that appeared in the August 2025 issue of the Notices of the AMS.
math.HO↗
arXiv subjects
James W. Anderson
Publications and source records attributed to James W. Anderson.
Omnipersistent signatures
In this note, we lay the groundwork for a new approach to the problem of group-signature classification of group actions on closed Riemann surfaces. This new approach first focuses on analyzing the low level arithmetic conditions on signatures before invoking the more complicated group theory. We provide the complete first step in this approach by giving the complete list of signatures which arithmetically could appear as a signature in every possible genus, and the subset of those which do appear as the signature of a group action in every possible genus.
math.GT↗
Prising apart geodesics by length in hyperbolic 3-manifolds
In this note, we develop a condition on a closed curve on a surface or in a 3-manifold that implies that the curve has the property that its length function on the space of all hyperbolic structures on the surface or 3-manifold completely determines the curve. For an orientable surface $S$ of negative Euler characteristic, we extend the known result that simple curves have this property to curves with self-intersection number one (with one exceptional case on closed surfaces of genus two that we describe completely), while for hyperbolizable 3-manifolds, we show that curves freely homotopic to simple curves on $\partial M$ have this property.
math.GT↗
Strong convergence of Kleinian groups: the cracked eggshell
In this paper we give a complete description of the set of discrete faithful representations SH(M) uniformizing a compact, orientable, hyperbolizable 3-manifold M with incompressible boundary, equipped with the strong topology, with the description given in term of the end invariants of the quotient manifolds. As part of this description, we introduce coordinates on SH(M) that extend the usual Ahlfors-Bers coordinates. We use these coordinates to show the local connectivity of SH(M) and study the action of the modular group of M on SH(M).
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Relative shapes of thick subsets of moduli space
A closed hyperbolic surface of genus $g\ge 2$ can be decomposed into pairs of pants along shortest closed geodesics and if these curves are sufficiently short (and with lengths uniformly bounded away from 0), then the geometry of the surface is essentially determined by the combinatorics of the pants decomposition. These combinatorics are determined by a trivalent graph, so we call such surfaces {\em trivalent}. In this paper, in a first attempt to understand the "shape" of the subset $\ts$ of moduli space consisting of surfaces whose systoles fill, we compare it metrically, asymptotically in g, with the set $\tri$ of trivalent surfaces. As our main result, we find that the set $\ts \cap \tri$ is metrically "sparse" in $\ts$ (where we equip $\moduli$ with either the Thurston or the Teichmüller metric).
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A Lower Bound for the Number of Group Actions on a Compact Riemann Surface
We prove that the number of distinct group actions on compact Riemann surfaces of a fixed genus $σ\geq 2$ is at least quadratic in $σ$. We do this through the introduction of a coarse signature space, the space $\mathcal{K}_σ$ of {\em skeletal signatures} of group actions on compact Riemann surfaces of genus $σ$. We discuss the basic properties of $\mathcal{K}_σ$ and present a full conjectural description.
math.AG↗
Small filling sets of curves on a surface
We show that the asymptotic growth rate for the minimal cardinality of a set of simple closed curves on a closed surface of genus $g$ which fill and pairwise intersect at most $K\ge 1$ times is $2\sqrt{g}/\sqrt{K}$ as $g \to \infty$ . We then bound from below the cardinality of a filling set of systoles by $g/\log(g)$. This illustrates that the topological condition that a set of curves pairwise intersect at most once is quite far from the geometric condition that such a set of curves can arise as systoles.
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On Automorphism Groups of Networks
We consider the size and structure of the automorphism groups of a variety of empirical `real-world' networks and find that, in contrast to classical random graph models, many real-world networks are richly symmetric. We relate automorphism group structure to network topology and discuss generic forms of symmetry and their origin in real-world networks.
physics.soc-ph↗
Free subgroups of surface mapping class groups
We quantify the generation of free subgroups of surface mapping class groups by pseudo-Anosov mapping classes in terms of their translation distance and the distance between their axes. Our methods make reference to \teichmuller space only.
math.GR↗
A simple criterion for non-relative hyperbolicity and one-endedness of groups
We give a combinatorial criterion that implies both the non-strong relative hyperbolicity and the one-endedness of a finitely generated group. We use this to show that many important classes of groups do not admit a strong relatively hyperbolic group structure and have one end. Applications include surface mapping class groups, the Torelli group, (special) automorphism and outer automorphism groups of most free groups, and the three-dimensional Heisenberg group. Our final application is to Thompson's group F.
math.GT↗
Uniformly exponential growth and mapping class groups of surfaces
We show that the mapping class group of an orientable finite type surface has uniformly exponential growth, as well as various closely related groups. This provides further evidence that mapping class groups may be linear.
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Conformal measures associated to ends of hyperbolic n-manifolds
Let Gamma be a non-elementary Kleinian group acting on the closed n-dimensional unit ball and assume that its Poincare series converges at the exponent alpha. Let M_Gamma be the Gamma-quotient of the open unit ball. We consider certain families E = {E_1,...,E_p} of open subsets of M_Gamma such that M_Gamma minus the union of all E_i is compact. The sets E_i are called ends of M_Gamma and E is called a complete collection of ends for M_Gamma. We show that we can associate to each end in E a conformal measure of dimension alpha such that the two measures corresponding to different ends are mutually singular if non-trivial. Each conformal measure for Gamma of dimension alpha on the limit set Lambda(Gamma) of Gamma can be written as a sum of such conformal measures associated to ends in E. In dimension 3, our results overlap with some results of Bishop and Jones.
math.CV↗
The minimal entropy problem for 3-manifolds with zero simplicial volume
We consider the minimal entropy problem, namely the question of whether there exists a smooth metric of minimal entropy, for certain classes of 3-manifolds. Among other resulsts, we show that if M is a closed, orientable, geometrizable 3-manifold with zero simplicial volume, then the minimal entropy can be solved for M if and only if M admits a metric modelled on 4 of the 8 standard 3-dimensional geometries, namely $S^3$, $S^2\times R$, $E^3$, or Nil.
math.DS↗
The topology of deformation spaces of Kleinian groups
Let M be a compact, hyperbolizable 3-manifold with nonempty incompressible boundary and let AH(π_1(M)) denote the space of (conjugacy classes of) discrete faithful representations of π_1(M) into PSL 2 (C). The components of the interior MP(π_1(M)) of AH(π_1(M)) (as a subset of the appropriate representation variety) are enumerated by the space A(M) of marked homeomorphism types of oriented, compact, irreducible 3-manifolds homotopy equivalent to M. In this paper, we give a topological enumeration of the components of the closure of MP(π_1(M)) and hence a conjectural topological enumeration of the components of AH(π_1(M)). We do so by characterizing exactly which changes of marked homeomorphism type can occur in the algebraic limit of a sequence of isomorphic freely indecomposable Kleinian groups. We use this enumeration to exhibit manifolds M for which AH(π_1(M)) has infinitely many components.
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Commensurability and locally free Kleinian groups
We show that there exist infinitely many commensurability classes of finite volume hyperbolic 3-manifolds whose fundamental group contains a subgroup which is locally free but not free. The main technical tool is the fact that a collection of hyperbolic 3-manifolds of bounded volume contains infinitely many commensurability classes. The result then by an application of Thurston's hyperbolization theorem for Haken 3-manifolds.
math.GT↗
The visual core of a hyperbolic 3-manifold
We introduce the notion of the visual core of a hyperbolic 3-manifold N and explore its basic properties. The visual core can be thought of as a harmonic analysis analogue of the convex core. We investigate circumstances in which the visual core of a cover N' of N embeds under the covering map from N' to N. We apply this analysis to convergent sequences of Kleinian groups, in order to understand when the visual core of the algebraic limit manifold embeds in the geometric limit manifold. We close with a discussion of the behavior of the visual core under Klein-Maskit combination.
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Cores of hyperbolic 3-manifolds and limits of Kleinian groups II
Troels Jorgensen conjectured that the algebraic and geometric limits of an algebraically convergent sequence of isomorphic Kleinian groups agree if there are no new parabolics in the algebraic limit. We prove that this conjecture holds in 'most' cases. In particular, we show that it holds when the domain of discontinuity of the algebraic limit of such a sequence is non-empty. We further show, with the same assumptions, that the limit sets of the groups in the sequence converge to the limit set of the algebraic limit. As a corollary, we verify the conjecture for finitely generated Kleinian groups which are not (non-trivial) free products of surface groups and infinite cyclic groups.
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A brief survey of the deformation theory of Kleinian groups
We give a brief overview of the current state of the study of the deformation theory of Kleinian groups. The topics covered include the definition of the deformation space of a Kleinian group and of several important subspaces; a discussion of the parametrization by topological data of the components of the closure of the deformation space; the relationship between algebraic and geometric limits of sequences of Kleinian groups; and the behavior of several geometrically and analytically interesting functions on the deformation space.
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