Maximal Representations Research Articles

Unstable minimal surfaces in $\mathbb{R}^{n}$ and in products of hyperbolic surfaces

We prove that every unstable equivariant minimal surface in \mathbb{R}^{n} produces a maximal representation of a surface group into \prod_{i=1}^{n}\mathrm{PSL}(2,\mathbb{R}) together with an unstable minimal surface in the corresponding product of closed hyperbolic surfaces. To do so, we lift the surface in \mathbb{R}^{n} to a surface in a product of \mathbb{R} -trees, then deform to a surface in a product of closed hyperbolic surfaces. We show that instability in one context implies instability in the other two.

Noncommutative Coordinates for Symplectic Representations

We introduce coordinates on the spaces of framed and decorated representations of the fundamental group of a surface with nonempty boundary into the symplectic group S p ( 2 n , R ) Sp(2n,\mathbf R) . These coordinates provide a noncommutative generalization of the parametrizations of the spaces of representations into S L ( 2 , R ) SL(2,\mathbf R) or P S L ( 2 , R ) PSL(2,\mathbf R) given by Thurston, Penner, Kashaev and Fock–Goncharov. On the space of decorated symplectic representations the coordinates give a geometric realization of the noncommutative cluster-like structures introduced by Berenstein–Retakh. The locus of positive coordinates maps to the space of framed maximal representations. We use this to determine an explicit homeomorphism between the space of framed maximal representations and a quotient by the group O ( n ) O(n) . This allows us to describe the homotopy type and, when n = 2 n=2 , to give an exact description of the singularities. Along the way, we establish a complete classification of pairs of nondegenerate quadratic forms.

Milnor–Wood inequality for klt varieties of general type and uniformization

AbstractWe generalize the definition of the Toledo invariant for representations of fundamental groups of smooth varieties of general type due to Koziarz and Maubon to the context of singular klt varieties, where the natural fundamental groups to consider are those of smooth loci. Assuming that the rank of the target Lie group is not greater than two, we show that the Toledo invariant satisfies a Milnor–Wood‐type inequality and we characterize the corresponding maximal representations.

Positive crossratios, barycenters, trees and applications to maximal representations

We study metric properties of maximal framed representations of fundamental groups of surfaces in symplectic groups over real closed fields, interpreted as actions on Bruhat–Tits buildings endowed with adapted Finsler norms. We prove that the translation length can be computed as intersection with a geodesic current, give sufficient conditions guaranteeing that such a current is a multicurve, and, if the current is a measured lamination, construct an isometric embedding of the associated tree in the building. These results are obtained as application of more general results of independent interest on positive crossratios and actions with compatible barycenters.

Weighted $$L^2$$ Holomorphic Functions on Ball-Fiber Bundles Over Compact Kähler Manifolds

Let $${\widetilde{M}}$$ be a complex manifold, $$\Gamma $$ be a torsion-free cocompact lattice of $$\text {Aut}({\widetilde{M}})$$ and $$\rho :\Gamma \rightarrow SU(N,1)$$ be a representation. Suppose that there exists a $$\rho $$ -equivariant totally geodesic isometric holomorphic embedding $$\imath :{{\widetilde{M}}}\rightarrow {\mathbb {B}}^N$$ . Let $$M:={{\widetilde{M}}}/\Gamma $$ and $$\Sigma :={\mathbb {B}}^N/\rho (\Gamma )$$ . In this paper, we investigate a relation between weighted $$L^2$$ holomorphic functions on the fiber bundle $$\Omega :=M\times _\rho {\mathbb {B}}^N$$ and the holomorphic sections of the pull-back bundle $$\imath ^*(S^mT^*_\Sigma )$$ over M. In particular, $$A^2_\alpha (\Omega )$$ has infinite dimension for any $$\alpha >-1$$ and if $$n<N$$ , then $$A^2_{-1}(\Omega )$$ also has the same property. As an application, if $$\Gamma $$ is a torsion-free cocompact lattice in SU(n, 1), $$n\ge 2$$ , and $$\rho :\Gamma \rightarrow SU(N,1)$$ is a maximal representation, then for any $$\alpha >-1$$ , $$A^2_\alpha ({\mathbb {B}}^n\times _{\rho } {\mathbb {B}}^N)$$ has infinite dimension. If $$n<N$$ , then $$A_{-1}^2({\mathbb {B}}^n\times _{\rho } \mathbb B^N)$$ also has the same property.

Arakelov–Milnor inequalities and maximal variations of Hodge structure

In this paper we study the $\mathbb {C}^*$-fixed points in moduli spaces of Higgs bundles over a compact Riemann surface for a complex semisimple Lie group and its real forms. These fixed points are called Hodge bundles and correspond to complex variations of Hodge structure. We introduce a topological invariant for Hodge bundles that generalizes the Toledo invariant appearing for Hermitian Lie groups. An important result of this paper is a bound on this invariant which generalizes the Milnor–Wood inequality for a Hodge bundle in the Hermitian case, and is analogous to the Arakelov inequalities of classical variations of Hodge structure. When the generalized Toledo invariant is maximal, we establish rigidity results for the associated variations of Hodge structure which generalize known rigidity results for maximal Higgs bundles and their associated maximal representations in the Hermitian case.

Application of the child community health inclusion index for measuring health inclusion of children with disabilities in the community: a feasibility study

BackgroundParticipation in the community is a fundamental human right for children with disabilities and is a key component of their health and development. Inclusive communities can enable children with disabilities to participate fully and effectively. The Child Community Health Inclusion Index (CHILD-CHII) is a comprehensive assessment tool developed to examine the extent to which community environments foster healthy, active living for children with disabilities.ObjectivesTo assess the feasibility of applying the CHILD-CHII measurement tool across different community settings.MethodsParticipants recruited through maximal representation, and purposeful sampling from four community sectors (Health, Education, Public Spaces, Community Organizations) applied the tool on their affiliated community facility. Feasibility was examined by assessing length, difficulty, clarity, and value for measuring inclusion; each rated on a 5-point Likert scale. Participants provided comments for each indicator through the questionnaire and a follow-up interview.ResultsOf the 12 participants, 92% indicated that the tool was ‘long’ or ‘much too long’; 66% indicated that the tool was clear; 58% indicated that the tool was ‘valuable’ or ‘very valuable’. No clear consensus was obtained for the level of difficulty. Participants provided comments for each indicator.ConclusionAlthough the length of the tool was regarded as long, it was seen to be comprehensive and valuable for stakeholders in addressing the inclusion of children with disabilities in the community. The perceived value and the evaluators’ knowledge, familiarity, and access to information can facilitate use of the CHILD-CHII. Further refinement and psychometric testing will be conducted.

Boundary maps and maximal representations on infinite-dimensional Hermitian symmetric spaces

AbstractWe define a Toledo number for actions of surface groups and complex hyperbolic lattices on infinite-dimensional Hermitian symmetric spaces, which allows us to define maximal representations. When the target is not of tube type, we show that there cannot be Zariski-dense maximal representations, and whenever the existence of a boundary map can be guaranteed, the representation preserves a finite-dimensional totally geodesic subspace on which the action is maximal. In the opposite direction, we construct examples of geometrically dense maximal representation in the infinite-dimensional Hermitian symmetric space of tube type and finite rank. Our approach is based on the study of boundary maps, which we are able to construct in low ranks or under some suitable Zariski density assumption, circumventing the lack of local compactness in the infinite-dimensional setting.

Multiplicative constants and maximal measurable cocycles in bounded cohomology

AbstractMultiplicative constants are a fundamental tool in the study of maximal representations. In this paper, we show how to extend such notion, and the associated framework, to measurable cocycles theory. As an application of this approach, we define and study the Cartan invariant for measurable$\mathrm{PU}(m,1)$-cocycles of complex hyperbolic lattices.

A MATSUMOTO–MOSTOW RESULT FOR ZIMMER’S COCYCLES OF HYPERBOLIC LATTICES

Following the philosophy behind the theory of maximal representations, we introduce the volume of a Zimmer’s cocycle Γ × X → PO° (n, 1), where Γ is a torsion-free (non-)uniform lattice in PO° (n, 1), with n > 3, and X is a suitable standard Borel probability Γ-space. Our numerical invariant extends the volume of representations for (non-)uniform lattices to measurable cocycles and in the uniform setting it agrees with the generalized version of the Euler number of self-couplings. We prove that our volume of cocycles satisfies a Milnor–Wood type inequality in terms of the volume of the manifold Γ\\ℍn. Additionally this invariant can be interpreted as a suitable multiplicative constant between bounded cohomology classes. This allows us to define a family of measurable cocycles with vanishing volume. The same interpretation enables us to characterize maximal cocycles for being cohomologous to the cocycle induced by the standard lattice embedding via a measurable map X → PO° (n, 1) with essentially constant sign.As a by-product of our rigidity result for the volume of cocycles, we give a different proof of the mapping degree theorem. This allows us to provide a complete characterization of maps homotopic to local isometries between closed hyperbolic manifolds in terms of maximal cocycles.In dimension n = 2, we introduce the notion of Euler number of measurable cocycles associated to a closed surface group and we show that it extends the classic Euler number of representations. Our Euler number agrees with the generalized version of the Euler number of self-couplings up to a multiplicative constant. Imitating the techniques developed in the case of the volume, we show a Milnor–Wood type inequality whose upper bound is given by the modulus of the Euler characteristic of the associated closed surface. This gives an alternative proof of the same result for the generalized version of the Euler number of self-couplings. Finally, using the interpretation of the Euler number as a multiplicative constant between bounded cohomology classes, we characterize maximal cocycles as those which are cohomologous to the one induced by a hyperbolization.

Basmajian-type inequalities for maximal representations

For suitable metrics on the locally symmetric space associated to a maximal representation, we prove inequalities between the length of the boundary and the lengths of orthogeodesics that generalize the classical Basmajian’s identity from Teichmuller theory. Any equality characterizes diagonal embeddings.

One Orthography, Four Lects: The Unified Berawan Orthography

Berawan is an endangered Austronesian language family consisting of four lects, which are Batu Belah, Long Teru, Long Jegan, and Long Terawan. Their settlements are located in the Malaysian state of Sarawak. The impetus for a unified orthography came from the Berawan community, who desire to write their lects consistently and reflecting the way they speak. The unified orthography was developed starting with a phonological analysis of the Berawan lects. This was followed by several orthography workshops and discussions with individual Berawan communities, culminating in a combined orthography workshop in which a unified orthography was agreed upon. The aim of the paper is to provide the groundwork for establishing the unifed orthography of the Berawan language family. A phonological comparison of the four Berawan varieties is included for this purpose. The phonological descriptions are taken from Burkhardt (2014). Smalley’s (1959, 1965) maxim of ‘maximal representation of speech’ and Rogers’s (2005) ‘shallow orthography’ approach are employed. On this basis, issues that arise for graphemic representation of Berawan phonemes are then discussed and the decisions made by the participants of the combined workshops are described. The paper also touches on issues encountered throughout the discussion. The issues that arose are primarily related to the differences in orthographic systems between the Berawan lects and the Malay language. The paper ends with a proposed unified Berawan orthography

Pieri Type Rules and GL(2|2) Tensor Products

We derive a closed formula for the tensor product of a family of mixed tensors using Deligne’s interpolating category underline {Rep}(GL_{0}). We use this formula to compute the tensor product of a family of irreducible GL(n|n)-representations. This includes the tensor product of any two maximal atypical irreducible representations of GL(2|2).

The geometry of maximal representations of surface groups into SO0(2,n)

In this paper, we study the geometric and dynamical properties of maximal representations of surface groups into Hermitian Lie groups of rank 2. Combining tools from Higgs bundle theory, the theory of Anosov representations, and pseudo-Riemannian geometry, we obtain various results of interest. We prove that these representations are holonomies of certain geometric structures, recovering results of Guichard and Wienhard. We also prove that their length spectrum is uniformly bigger than that of a suitably chosen Fuchsian representation, extending a previous work of the second author. Finally, we show that these representations preserve a unique minimal surface in the symmetric space, extending a theorem of Labourie for Hitchin representations in rank 2.

Maximal representations of cocompact complex hyperbolic lattices, a uniform approach

We complete the classification of maximal representations of cocompact complex hyperbolic lattices in Hermitian Lie groups by dealing with the exceptional groups E 6(-14) and E 7(-25) . We prove that if ρ is a maximal representation of a cocompact complex hyperbolic lattice Γ⊂SU(1,n), n>1, in an exceptional Hermitian group G ℝ , then n=2 and G ℝ =E 6(-14) , and we describe completely the representation ρ. The case of classical Hermitian target groups was treated by Vincent Koziarz and the second named author [KM17]. However we do not focus immediately on the exceptional cases and instead we provide a more unified perspective, as independent as possible of the classification of the simple Hermitian Lie groups. This relies on the study of the cominuscule representation of the complexification G of the target group G ℝ . As a by-product of our methods, when the target Hermitian group G ℝ has tube type, we obtain an inequality on the Toledo invariant of the representation ρ:Γ→G ℝ which is stronger than the Milnor-Wood inequality (thereby excluding maximal representations in such groups).

Positively ratioed representations

Let S be a closed orientable surface of genus at least 2 and let G be a semisimple real algebraic group of non-compact type. We consider a class of representations from the fundamental group of S to G called positively ratioed representations. These are Anosov representations with the additional condition that certain associated cross ratios satisfy a positivity property. Examples of such representations include Hitchin representations and maximal representations. Using geodesic currents, we show that the corresponding length functions for these positively ratioed representations are well-behaved. In particular, we prove a systolic inequality that holds for all such positively ratioed representations.

Survey Response Rate and Quality in a Mental Health Clinic Population: Results from a Randomized Survey Comparison.

Given the limited ability of informatics-based assessment technologies to reach individuals with serious mental health conditions, this study evaluated the feasibility and data quality of mail-out healthcare surveys in this population to assist in measure selection for a multi-site controlled implementation trial. Veterans were randomly selected from those who had received services at a mental health clinic in the Department of Veterans Affairs, and were randomly assigned to one of three questionnaire lengths. Survey length (48-127 items) was not associated with differences in response rate, percent of items missing, or data quality. However, internal consistency reliability was variable among scales and survey lengths. Additional analyses indicate the above measures of survey data quality may differ among respondents who are non-white and younger and have psychotic disorders. These results can inform survey protocols to ensure maximal representation of this vulnerable population in health planning and policy assessment.

Exotic components of SO(p,q) surface group representations, and their Higgs bundle avatars

For semisimple Lie groups, moduli spaces of Higgs bundles on a Riemann surface correspond to representation varieties for the surface fundamental group. In many cases, natural topological invariants label connected components of the moduli spaces. Hitchin representations into split real forms, and maximal representations into Hermitian Lie groups, are the only previously know cases where natural invariants do not fully distinguish connected components. In this note we announce the existence of new such exotic components in the moduli spaces for the groups SO(p,q) with 2<p<q. These groups lie outside formerly know classes of groups associated with exotic components.

Schottky groups and maximal representations

We describe a construction of Schottky type subgroups of automorphism groups of partially cyclically ordered sets. We apply this construction to the Shilov boundary of a Hermitian symmetric space and show that in this setting Schottky subgroups correspond to maximal representations of fundamental groups of surfaces with boundary. As an application, we construct explicit fundamental domains for the action of maximal representations into $\mathrm{Sp}(2n,\mathbb{R})$ on $\mathbb{RP}^{2n-1}$.

Maximal representations, non-Archimedean Siegel spaces, and buildings

Let $\mathbb F$ be a real closed field. We define the notion of a maximal framing for a representation of the fundamental group of a surface with values in ${\rm Sp}(2n,\mathbb F)$. We show that ultralimits of maximal representations in ${\rm Sp}(2n,\mathbb R)$ admit such a framing, and that all maximal framed representations satisfy a suitable generalisation of the classical Collar Lemma. In particular this establishes a Collar Lemma for all maximal representations into ${\rm Sp}(2n,\mathbb R)$. We then describe a procedure to get from representations in ${\rm Sp}(2n,\mathbb F)$ interesting actions on affine buildings, and, in the case of representations admitting a maximal framing, we describe the structure of the elements of the group acting with zero translation length.