Twisted dynamical zeta funcions and the Fried's conjeture

Abstract: This is a survey article on the twisted dynamical zeta functions of Ruelle and Selberg and the Fried's conjecture. It is based on the mini-course: "Twisted Ruelle zeta function, complex-valued analytic torsion and the Fried's conjecture", given by the author during the thematic trimester programme: "Representation Theory and Noncommutative Geometry" at Institut Henri Poincaré.

• The paper assembles results on compact hyperbolic surfaces, orbisurfaces, and odd-dimensional manifolds, using a unified approach
• The survey shows that twisted Ruelle and Selberg zeta functions exhibit meromorphic continuation and functional equations, connecting to topological invariants like Reidemeister–Turaev and Cappell–Miller torsions for certain representations.
• Non-unitary twists introduce challenges, but the study identifies convergence bounds and uses Müller's Selberg trace formula for non-self-adjoint Laplacians to achieve these results.

This paper is a survey by Polyxeni Spilioti of twisted Ruelle and Selberg zeta functions and their relation to Fried's conjecture, based on a mini-course given at Institut Henri Poincaré during the thematic trimester "Representation Theory and Noncommutative Geometry" (2603.03156). It assembles the author's own results — on compact hyperbolic surfaces, compact hyperbolic orbisurfaces, and odd-dimensional compact hyperbolic manifolds with non-unitary twists — into a unified account organized around a common technique: the Selberg trace formula for non-self-adjoint twisted Laplacians.

Dynamical zeta functions

For a compact hyperbolic manifold $X$, the Ruelle zeta function is defined as an Euler-type product over prime closed geodesics $\gamma$ of length $\ell(\gamma)$:

$$R(s)=\prod_{\gamma\ \mathrm{prime}}\bigl(1-e^{-s\ell(\gamma)}\bigr),$$

while the Selberg zeta function is the double product

$$Z(s)=\prod_{\gamma\ \mathrm{prime}}\prod_{k=0}^{\infty}\det\bigl(1-e^{-(s+k)\ell(\gamma)}\bigr),$$

both converging for $\Re(s)$ large. The analogy with the Riemann zeta function is made precise via the correspondence between primes $p$ and quantities $e^{\ell(\gamma)}$. The Selberg trace formula yields the meromorphic continuation of $Z(s)$ to all of $\mathbb{C}$, with spectral zeros on the line $\gamma$0, the analogue of the critical line in number theory (2603.03156). For hyperbolic surfaces the two functions are related simply by $\gamma$1, so analytic properties of $\gamma$2 follow from those of $\gamma$3.

In general rank-one locally symmetric settings, $\gamma$4 with $\gamma$5 a quotient of $\gamma$6, $\gamma$7, $\gamma$8, or $\gamma$9, Bunke–Olbrich define twisted versions associated with finite-dimensional unitary representations $\ell(\gamma)$0 of $\ell(\gamma)$1 and $\ell(\gamma)$2 of $\ell(\gamma)$3, with products over prime conjugacy classes involving $\ell(\gamma)$4.

Fried's conjecture

Fried's conjecture asks whether special values of the twisted Ruelle zeta function at zero equal a topological invariant. Fried himself settled the case of orthogonal (acyclic) representations: if $\ell(\gamma)$5 is acyclic, then $\ell(\gamma)$6 extends meromorphically to $\ell(\gamma)$7 and

$\ell(\gamma)$8

the Reidemeister torsion of the unit tangent bundle (2603.03156). The survey then records the resolution for arbitrary complex representations in three settings due to Frahm–Spilioti, Bénard–Frahm–Spilioti, and Spilioti:

• Compact hyperbolic surfaces: $\ell(\gamma)$9 admits holomorphic continuation with an explicit zero product determined by the spectrum of the non-self-adjoint twisted Laplacian $$R(s)=\prod_{\gamma\ \mathrm{prime}}\bigl(1-e^{-s\ell(\gamma)}\bigr),$$0; consequently $$R(s)=\prod_{\gamma\ \mathrm{prime}}\bigl(1-e^{-s\ell(\gamma)}\bigr),$$1, and near zero

$$R(s)=\prod_{\gamma\ \mathrm{prime}}\bigl(1-e^{-s\ell(\gamma)}\bigr),$$2

with the sign equal to $$R(s)=\prod_{\gamma\ \mathrm{prime}}\bigl(1-e^{-s\ell(\gamma)}\bigr),$$3 where $$R(s)=\prod_{\gamma\ \mathrm{prime}}\bigl(1-e^{-s\ell(\gamma)}\bigr),$$4 is the multiplicity of the eigenvalue $$R(s)=\prod_{\gamma\ \mathrm{prime}}\bigl(1-e^{-s\ell(\gamma)}\bigr),$$5 of $$R(s)=\prod_{\gamma\ \mathrm{prime}}\bigl(1-e^{-s\ell(\gamma)}\bigr),$$6.

• Compact hyperbolic orbisurfaces: for irreducible $$R(s)=\prod_{\gamma\ \mathrm{prime}}\bigl(1-e^{-s\ell(\gamma)}\bigr),$$7, if $$R(s)=\prod_{\gamma\ \mathrm{prime}}\bigl(1-e^{-s\ell(\gamma)}\bigr),$$8 (where $$R(s)=\prod_{\gamma\ \mathrm{prime}}\bigl(1-e^{-s\ell(\gamma)}\bigr),$$9 generates the generic fiber of the Seifert fibration), the order of vanishing at $$Z(s)=\prod_{\gamma\ \mathrm{prime}}\prod_{k=0}^{\infty}\det\bigl(1-e^{-(s+k)\ell(\gamma)}\bigr),$$0 is $$Z(s)=\prod_{\gamma\ \mathrm{prime}}\prod_{k=0}^{\infty}\det\bigl(1-e^{-(s+k)\ell(\gamma)}\bigr),$$1; if $$Z(s)=\prod_{\gamma\ \mathrm{prime}}\prod_{k=0}^{\infty}\det\bigl(1-e^{-(s+k)\ell(\gamma)}\bigr),$$2, then $$Z(s)=\prod_{\gamma\ \mathrm{prime}}\prod_{k=0}^{\infty}\det\bigl(1-e^{-(s+k)\ell(\gamma)}\bigr),$$3 is acyclic and $$Z(s)=\prod_{\gamma\ \mathrm{prime}}\prod_{k=0}^{\infty}\det\bigl(1-e^{-(s+k)\ell(\gamma)}\bigr),$$4, the Reidemeister–Turaev torsion in the Euler structure induced by the geodesic flow.
• Odd-dimensional compact hyperbolic manifolds: $$Z(s)=\prod_{\gamma\ \mathrm{prime}}\prod_{k=0}^{\infty}\det\bigl(1-e^{-(s+k)\ell(\gamma)}\bigr),$$5 continues meromorphically for every $$Z(s)=\prod_{\gamma\ \mathrm{prime}}\prod_{k=0}^{\infty}\det\bigl(1-e^{-(s+k)\ell(\gamma)}\bigr),$$6, and a determinant formula expresses $$Z(s)=\prod_{\gamma\ \mathrm{prime}}\prod_{k=0}^{\infty}\det\bigl(1-e^{-(s+k)\ell(\gamma)}\bigr),$$7 as products of graded determinants of shifted flat Hodge Laplacians together with an explicit exponential correction involving $$Z(s)=\prod_{\gamma\ \mathrm{prime}}\prod_{k=0}^{\infty}\det\bigl(1-e^{-(s+k)\ell(\gamma)}\bigr),$$8.

The vanishing order in the odd-dimensional case equals $$Z(s)=\prod_{\gamma\ \mathrm{prime}}\prod_{k=0}^{\infty}\det\bigl(1-e^{-(s+k)\ell(\gamma)}\bigr),$$9 where $\Re(s)$0; this connects the order of vanishing directly to Betti numbers when Hodge theory applies.

Non-unitary twists

The central technical difficulty addressed in the survey is that for arbitrary (non-unitary) $\Re(s)$1, the Euler products no longer converge in the usual half-plane. The remedy is a growth bound on traces along conjugacy classes: there exist $\Re(s)$2 such that $\Re(s)$3 for all $\Re(s)$4 (2603.03156). Combined with the standard counting estimate $\Re(s)$5, this gives convergence in some right half-plane, with constants $\Re(s)$6 that can be made explicit. In dimension two, the bound follows from quasi-isometry of word length and hyperbolic distance; the paper notes that the dependence on the chosen norm can be removed via the critical exponent $\Re(s)$7, a notion also used by Naud and the author to study the bottom of the spectrum of twisted Laplacians. For orbisurfaces, convergence follows from Fedosova–Pohl.

A structural limitation appears already here and persists throughout: since the twisted Laplacian $\Re(s)$8 is not self-adjoint, Hodge theory fails — one has $\Re(s)$9 in general — so cohomological vanishing cannot be read off from spectral data.

Trace formulas for non-unitary twists

The engine behind all continuation results is Müller's Selberg trace formula for non-unitary twists, applied to the twisted Bochner Laplacian $p$0, which shares its principal symbol with the self-adjoint $p$1 and hence has discrete spectrum in a translate of a cone containing $p$2. The heat operator $p$3 has smooth kernel and is trace class; Lidskii's theorem identifies its trace with $p$4 over generalized eigenspaces. The geometrical side decomposes into identity and hyperbolic contributions, with the latter involving the Harish-Chandra Schwartz space membership of the heat kernel and the spherical Fourier transform identity $p$5 on surfaces (2603.03156).

The survey states three trace formulas explicitly:

Setting	Twisted operator	Geometric side features
Compact hyperbolic surface	$p$6	Hyperbolic classes only
Orbisurface	$p$7	Hyperbolic and elliptic classes; Plancherel denominator $p$8
Odd-dimensional hyperbolic $p$9	$e^{\ell(\gamma)}$0, $e^{\ell(\gamma)}$1	Weyl-invariant vs. non-invariant cases differ by factor 2

For orbisurfaces, the elliptic contributions carry terms depending on the rotation angles $e^{\ell(\gamma)}$2 and the character $e^{\ell(\gamma)}$3 of the fiber; these are precisely the contributions later responsible for the torsion factor appearing in determinant formulas.

Meromorphic continuation and functional equations

On surfaces, the route is via a resolvent trace formula. Shifting $e^{\ell(\gamma)}$4, taking differences of resolvents (trace class by Weyl's law), integrating against the geometric side of the trace formula isolates the logarithmic derivative $e^{\ell(\gamma)}$5:

$e^{\ell(\gamma)}$6

The resulting resolvent trace formula shows $e^{\ell(\gamma)}$7 extends meromorphically with poles at $e^{\ell(\gamma)}$8 (spectral zeros, counted with algebraic multiplicity of eigenvalues $e^{\ell(\gamma)}$9 of $Z(s)$0) plus trivial-series poles at negative integers with residues forced to be positive integers by Gauss–Bonnet, $Z(s)$1 (2603.03156). Integration and exponentiation give the holomorphic continuation of $Z(s)$2 with the stated zero product, and the identity $Z(s)$3 gives meromorphic continuation of $Z(s)$4. A notable consequence: the functional equation $Z(s)$5 immediately forces the exact leading behavior at zero, $Z(s)$6, so the order of vanishing is purely topological, proportional to the Euler characteristic.

The same mechanism — identifying the hyperbolic contribution of the trace formula with the logarithmic derivative of the twisted Selberg zeta function — works verbatim for orbisurfaces and odd-dimensional manifolds, though the representation-theoretic bookkeeping is heavier.

Determinant formulas, torsion factors, and analytic torsion

For odd-dimensional compact hyperbolic manifolds, Spilioti derives a determinant formula representing $Z(s)$7 as a product of graded regularized determinants of operators $Z(s)$8 times an exponential correction term (2603.03156). Since acyclicity does not force $Z(s)$9 without Hodge theory, regularity of $\mathbb{C}$0 at zero is obtained only for representations in a neighborhood $\mathbb{C}$1 (in classical topology) of the acyclic unitary locus on which the odd signature operator $\mathbb{C}$2 is bijective, using a continuity argument. Under this assumption,

$\mathbb{C}$3

the complex-valued Cappell–Miller torsion. This is the precise sense in which Fried's conjecture holds here: the special value is a spectral rather than topological invariant, and equality with refined analytic torsion (Braverman–Kappeler) follows as well. The restriction to a neighborhood of the unitary acyclic locus is an explicit hypothesis, not a theorem for all acyclic $\mathbb{C}$4.

The survey also reports joint work with Jorgenson and Smajlović proving a determinant formula for twisted Selberg zeta functions on orbisurfaces:

$\mathbb{C}$5

where the torsion factor $\mathbb{C}$6 carries an explicit contribution from each elliptic class of order $\mathbb{C}$7, weighted by $\mathbb{C}$8 and the traces $\mathbb{C}$9. Two consequences are highlighted: if $\gamma$00 is not an eigenvalue of $\gamma$01, then $\gamma$02 and the regularized determinant at zero equals the product of the three zeta factors at $\gamma$03 times $\gamma$04; and along Yamaguchi's families of non-unitary representations, the normalized torsion factor satisfies

$\gamma$05

matching the asymptotics of higher-dimensional Reidemeister torsion on the unit tangent bundle up to an absolute constant (2603.03156).

Related work and limitations

The survey situates these results within a broad literature: Bunke–Olbrich proved Fried's conjecture for unitary twists on all real rank-one locally symmetric spaces; higher-rank unitary cases were handled by Moscovici–Stanton(-Frahm) and Shen; microlocal methods appear in Dyatlov–Zworski, Dang–Guillarmou–Rivière–Shen, Chaubet–Dang, and Cekić–Delarue–Dyatlov–Paternain; equivariant extensions are due to Hochs–Saratchandran and Hochs–Pirie; related arithmetic work includes Abdurrahman–Venkatesh on symplectic L-functions and Reidemeister torsion mod squares.

Several limitations are stated plainly within the paper's own framework. First, the special-value theorems in the odd-dimensional case hold only for representations close to unitary ones; the behavior of $\gamma$06 at zero for general acyclic non-unitary $\gamma$07 remains unresolved there. Second, on surfaces and orbisurfaces the results concern orders of vanishing and functional equations rather than equality with torsion except in the specific orbifold case $\gamma$08; the sign ambiguity is resolved only spectrally, via multiplicities of the zero eigenvalue of twisted Laplacians. Third, the determinant formula on orbisurfaces involves the extra factors $\gamma$09, $\gamma$10 and the torsion factor $\gamma$11, so it does not reduce to a clean equality between $\gamma$12 and a spectral determinant alone. Open questions left by the material include extending the Cappell–Miller torsion equality beyond the neighborhood $\gamma$13 of the acyclic unitary locus, and obtaining comparable special-value statements for even-dimensional hyperbolic manifolds and higher-rank spaces with non-unitary coefficients.

Conclusion

The survey presents a coherent methodological narrative: non-unitary trace formulas for twisted, non-self-adjoint Laplacians yield resolvent trace formulas; these produce meromorphic continuations, functional equations, and exact orders of vanishing for twisted Ruelle and Selberg zeta functions on compact hyperbolic surfaces, orbisurfaces, and odd-dimensional hyperbolic manifolds; and, under explicit hypotheses on the representation, the value at zero is identified with Reidemeister–Turaev or Cappell–Miller torsion, confirming Fried's conjecture in those regimes. The unification of these cases through a single trace-formula technique, applicable uniformly across rank-one locally symmetric spaces, is the paper's principal contribution as an exposition.