For any rank-one Riemannian symmetric space S of non-compact type and any discrete, cofinite, non-cocompact, torsion-free group Γ of orientation-preserving Riemannian isometries on S, we develop a cohomological interpretation for the cusp forms of Γ. To that end, we identify certain Γ-submodules of smooth semi-analytic vectors in the spherical principal series representation with spectral parameter ν as well as certain subspaces of parabolic cohomology spaces of Γ of degree dim S-1 with these Γ-submodules. We provide explicit isomorphisms between the spaces of cusp forms of spectral parameter ν and these specific cohomology subspaces. The isomorphisms from cusp forms to cohomology are given by an integral transform, and the explicit form of the inverse isomorphism takes advantage of a certain reproducing property of the integral transform. The result is uniform for all these symmetric spaces and does not rely on their classification.

For geometrically finite non-compact developable hyperbolic orbisurfaces (including those of infinite volume), we provide transfer operator families whose Fredholm determinants are identical to the Selberg zeta function. Our proof yields an algorithmic and uniform construction. This construction is initiated with an externally provided cross section for the geodesic flow on the considered orbisurface that yields a highly faithful, but non-uniformly expanding discrete dynamical system modelling the geodesic flow. Through a number of algorithmic steps of reduction, extension, translation, induction and acceleration, we turn this cross section into one that yields a still highly faithful, but now uniformly expanding discrete dynamical system. The arising transfer operator family is nuclear of order zero on suitable Banach spaces. In addition, finite-dimensional twists with non-expanding cusp monodromy can be included.

For the Selberg zeta function of geometrically finite infinite-area hyperbolic orbisurfaces with twists by finite-dimensional unitary representations, we establish a factorization formula in terms of a Weierstrass product of the Laplace resonances of the considered hyperbolic orbisurface, Barnes G-functions, gamma functions, and the singularity degrees of the representation. We thereby provide an interpretation of the zeros and poles of the Selberg zeta function by spectral and geometric entities of the orbisurface and the representation. This formula generalizes the factorization result by Borthwick, Judge and Perry to hyperbolic orbisurfaces with orbifold singularities as well as to unitary twists. Also in the untwisted case, the presence of orbifold singularities yields a separate, previously unobserved contribution to the factorization formula.

In recent years, significant advancements have been made in understanding the spectral theory of hyperbolic spaces, particularly in the context of twisting representations, both unitary as well as non-unitary. We survey some results that we obtained within the framework of the SPP 2026 "Geometry at Infinity" as well as some closely related results.

Cohomology theories have long proven to be powerful, unifying tools in numerous areas of mathematics, specifically in algebra, geometry, and number theory. Historically, the invention of the “right” cohomology theory often proved to be the key to uniform, conceptual proofs of conjectures and explanations of heuristic phenomena, bringing together seemingly unrelated mathematical areas. This workshop concentrated on the areas of automorphic forms and enumerative algebra, two fields in which cohomology theories already had a particular impact. It brought together mathematicians from various subdirections of these areas, both junior and seniorr

In the framework of infinite ergodic theory, we derive equidistribution results for suitable weighted sequences of cusp points of Hecke triangle groups encoded by group elements of constant word length with respect to a set of natural generators. This is a generalization of the corresponding results for the modular group, for which we rely on advanced results from infinite ergodic theory and transfer operator techniques developed for AFN-maps.

For vector-valued Maass cusp forms for SL_2(ℤ) with real weight k∈ℝ and spectral parameter s∈ℂ, Re s∈ (0,1), s≢± k/2 mod 1, we propose a notion of vector-valued period functions, and we establish a linear isomorphism between the spaces of Maass cusp forms and period functions by means of a cohomological approach. The period functions are a generalization of those for the classical Maass cusp forms, being solutions of a finite-term functional equation or, equivalently, eigenfunctions with eigenvalue 1 of a transfer operator deduced from the geodesic flow on the modular surface. We apply this result to deduce a notion of period functions and related linear isomorphism for Jacobi Maass forms of weight k+1/2 for the semi-direct product of SL_2(ℤ) with the integer points Hei(ℤ) of the Heisenberg group.

We present the Laplace operator associated to a hyperbolic surface Gamma\H and a unitary representation of the fundamental group, extending the previous definition for hyperbolic surfaces of finite area to those of infinite area. We show that the resolvent of this operator admits a meromorphic continuation to all of C by constructing a parametrix for the Laplacian, following the approach by Guillope and Zworski. We use the construction to provide an optimal upper bound for the counting function of the poles of the continued resolvent.

We develop cohomological interpretations for several types of automorphic forms for Hecke triangle groups of infinite covolume. We then use these interpretations to establish explicit isomorphisms between spaces of automorphic forms, cohomology spaces and spaces of eigenfunctions of transfer operators. These results show a deep relation between spectral entities of Hecke surfaces of infinite volume and the dynamics of their geodesic flows.

We provide Fourier expansions of vector-valued eigenfunctions of the hyperbolic Laplacian that are twist-periodic in a horocycle direction. The twist may be given by any endomorphism of a finite-dimensional vector space; no assumptions on invertibility or unitarity are made. Examples of such eigenfunctions include vector-valued twisted automorphic forms of Fuchsian groups. We further provide a detailed description of the Fourier coefficients and explicitly identify each of their constituents, which intimately depend on the eigenvalues of the twisting endomorphism and the size of its Jordan blocks. In addition, we determine the growth properties of the Fourier coefficients.

We study the spectral properties of the Laplace operator associated to a hyperbolic surface in the presence of a unitary representation of the fundamental group. Following the approach by Guillopé and Zworski, we establish a factorization formula for the twisted scattering determinant and describe the behavior of the scattering matrix in a neighborhood of 1/2.

We present a numerical method to calculate resonances of Schottky surfaces based on Selberg theory, transfer operator techniques and Lagrange-Chebyshev approximation. This method is an alternative to the method based on periodic orbit expansion used previously in this context.

We initiate the study of Selberg zeta functions $Z_{\Gamma,\chi}$ for geometrically finite Fuchsian groups $\Gamma$ and finite-dimensional representations $\chi$ with non-expanding cusp monodromy. We show that for all choices of $(\Gamma,\chi)$, the Selberg zeta function $Z_{\Gamma,\chi}$ converges on some half-plane in $\mathbb{C}$. In addition, under the assumption that $\Gamma$ admits a strict transfer operator approach, we show that $Z_{\Gamma,\chi}$ extends meromorphically to all of $\mathbb{C}$.

The correspondence principle in physics between quantum mechanics and classical mechanics suggests deep relations between spectral and geometric entities of Riemannian manifolds. We survey - in a way intended to be accessible to a wide audience of mathematicians - a mathematically rigorous instance of such a relation that emerged in recent years, showing a dynamical interpretation of certain Laplace eigenfunctions of hyperbolic surfaces.

The investigation of L-Laplace eigenvalues and eigenfunctions for hyperbolic surfaces of finite area is a classical and exciting topic at the intersection of number theory, harmonic analysis and mathematical physics. In stark contrast, for geometrically finite hyperbolic surfaces of infinite area, the discrete L-spectrum is finite. A natural replacement are the resonances of the considered hyperbolic surface, which are the poles of the meromorphically continued resolvent R(s) = (∆− s(1− s))−1

Let $$\Gamma $$ be a geometrically finite Fuchsian group and suppose that $$\chi :\Gamma \rightarrow {{\,\mathrm{GL}\,}}(V)$$ is a finite-dimensional representation with non-expanding cusp monodromy. We show that the parabolic Eisenstein series for $$\Gamma $$ with twist $$\chi $$ converges on some half-plane. Further, we develop Fourier-type expansions for these Eisenstein series.

We investigate how bounds of resonance counting functions for Schottky surfaces behave under transitions to covering surfaces of finite degree. We consider the classical resonance counting function asking for the number of resonances in large (and growing) disks centered at the origin of $\mathcal C$, as well as the (fractal) resonance counting function asking for the number of resonances in boxes near the axis of the critical exponent. For the former counting function we provide a transfer-operator-based proof that bounding constants can be chosen such that the transformation behavior under transition to covers is as for the Weyl law in the case of surfaces of finite area. For the latter counting function we deduce a bound in terms of the covering degree and the minimal length of a periodic geodesic on the covering surface. This yields an improved fractal Weyl upper bound. In the setting of Schottky surfaces, these estimates refine previous results due to Guillopé–Zworski and Guillopé–Lin–Zworski. When applied to principal congruence covers, these results yield new estimates for the resonance counting functions in the level aspect, which have recently been investigated by Jakobson–Naud. The techniques used in this article are based on the thermodynamic formalism for L-functions (twisted Selberg zeta functions), and twisted transfer operators.

We provide special cross sections for the Weyl chamber flow on a sample class of Riemannian locally symmetric spaces of higher rank, namely the direct product spaces of Schottky surfaces. We further present multi-parameter transfer operator families for the discrete dynamical systems on Furstenberg boundary that are related to these cross sections.

Over the last few years Pohl (partly jointly with coauthors) has developed dual ‘slow/fast’ transfer operator approaches to automorphic functions, resonances, and Selberg zeta functions for a certain class of hyperbolic surfaces $\unicode[STIX]{x1D6E4}\backslash \mathbb{H}$ with cusps and all finite-dimensional unitary representations $\unicode[STIX]{x1D712}$ of $\unicode[STIX]{x1D6E4}$ . The eigenfunctions with eigenvalue 1 of the fast transfer operators determine the zeros of the Selberg zeta function for $(\unicode[STIX]{x1D6E4},\unicode[STIX]{x1D712})$ . Further, if $\unicode[STIX]{x1D6E4}$ is cofinite and $\unicode[STIX]{x1D712}$ is the trivial one-dimensional representation then highly regular eigenfunctions with eigenvalue 1 of the slow transfer operators characterize Maass cusp forms for $\unicode[STIX]{x1D6E4}$ . Conjecturally, this characterization extends to more general automorphic functions as well as to residues at resonances. In this article we study, without relying on Selberg theory, the relation between the eigenspaces of these two types of transfer operators for any Hecke triangle surface $\unicode[STIX]{x1D6E4}\backslash \mathbb{H}$ of finite or infinite area and any finite-dimensional unitary representation $\unicode[STIX]{x1D712}$ of the Hecke triangle group $\unicode[STIX]{x1D6E4}$ . In particular, we provide explicit isomorphisms between relevant subspaces. This solves a conjecture by Möller and Pohl, characterizes some of the zeros of the Selberg zeta functions independently of the Selberg trace formula, and supports the previously mentioned conjectures.