Culler–Shalen Norm in 3-Manifolds

• The Culler–Shalen norm is a piecewise-linear seminorm on H₁(∂M; ℝ) defined via pole orders of peripheral trace functions, encoding detected boundary slopes.
• It bridges algebraic geometry and 3-manifold topology by linking valuations at ideal points in character varieties to the construction of essential surfaces.
• Applications include studying hyperbolic knot exteriors and A-polynomial Newton polygons, with robust formulations even in arbitrary characteristic.

Searching arXiv for recent and foundational papers on the Culler–Shalen norm and closely related character-variety constructions. The Culler–Shalen norm is a piecewise-linear seminorm, and in many cases a genuine norm, on $H_1(\partial M;\mathbb{R})$ associated to an irreducible curve in the $\mathrm{SL}_2$ character variety of a compact, orientable, irreducible $3$–manifold with torus boundary. It is defined from the pole behavior of peripheral trace functions at ideal points of the curve and, through the Culler–Shalen construction, encodes the boundary slopes of essential surfaces detected by those ideal points. In the classical complex setting and in the arbitrary-characteristic extension developed in "An invitation to Culler–Shalen theory in arbitrary characteristic" (Garden et al., 2024), the norm serves as a bridge between the algebraic geometry of character varieties, actions on Bass–Serre or Bruhat–Tits trees, and the topology of incompressible surfaces in $3$–manifolds. Closely related formulations express the same quantity through degrees of trace maps, weighted intersection numbers with detected boundary slopes, and valuations of peripheral eigenvalues (Garden et al., 2024).

1. Foundational setting and character-variety framework

Let $M$ be a compact, orientable, irreducible $3$–manifold with one torus boundary component $\partial M$, and let $\pi_1(M)$ denote its fundamental group. For an algebraically closed field $k$, the $\mathrm{SL}_2(k)$ representation variety is

$\mathrm{SL}_2$0

The associated variety of characters is denoted $\mathrm{SL}_2$1, and its coordinate ring is generated by trace functions $\mathrm{SL}_2$2 (Garden et al., 2024). In the complex case, one also writes

$\mathrm{SL}_2$3

with coordinate ring generated by trace functions

$\mathrm{SL}_2$4

(Kitayama, 2014).

Fix a basis $\mathrm{SL}_2$5 of $\mathrm{SL}_2$6 corresponding to meridian and longitude. A slope $\mathrm{SL}_2$7 is represented by $\mathrm{SL}_2$8. If $\mathrm{SL}_2$9 is an irreducible curve, then its smooth projective completion $3$0 contains finitely many ideal points $3$1 (Garden et al., 2024). Each ideal point determines a discrete rank–$3$2 valuation on the function field of the curve; this valuation is the basic local datum from which the norm is defined (Garden et al., 2024, Kitayama, 2014).

In this setting, the norm records how peripheral trace functions degenerate at the ideal points. The algebraic definition is local in valuation-theoretic terms, but the resulting function on $3$3 is global and polyhedral. The unit ball is a finite, centrally symmetric convex polygon, often called the Culler–Shalen norm polygon (Chesebro, 2012). Its supporting lines are determined by the valuations at the ideal points, and its edges encode detected boundary slopes (Chesebro, 2012).

2. Definition by trace degrees and valuations

For $3$4, the trace function $3$5 on $3$6 extends rationally to $3$7. The Culler–Shalen seminorm attached to $3$8 is defined by the degree of this rational function, equivalently by summing its pole orders at the ideal points:

$3$9

Since $3$0 is regular on $3$1, only ideal points contribute:

$3$2

This is the original Culler–Shalen seminorm definition via degrees and valuations on character curves (Garden et al., 2024). Equivalent formulations in the complex literature write

$3$3

where a standard normalization uses

$3$4

(Kitayama, 2014). Any normalization equivalent up to multiplication by a unit on $3$5 yields the same seminorm, since the degree only records poles at ideal points (Kitayama, 2014).

A directly comparable formulation used in the module-theoretic literature is

$3$6

for an irreducible curve component $3$7 (Chesebro, 2012). In the normalization based on $3$8 rather than $3$9, one has

$M$0

(Katerba, 2018).

These formulas express the seminorm as total polar degree. In particular, the seminorm is nonnegative and homogeneous, and the polygonal structure of its unit ball reflects the fact that the total pole order is assembled from finitely many valuations (Garden et al., 2024, Chesebro, 2012).

3. Ideal points, tree actions, and essential surfaces

The geometric meaning of the norm comes from the Culler–Shalen construction. If $M$1 is an ideal point of an irreducible curve $M$2, then after lifting $M$3 to a curve in the representation variety, the tautological representation

$M$4

yields an action of $M$5 on the Bruhat–Tits tree $M$6 of $M$7 over the valued field (Garden et al., 2024). In the complex setting this is the corresponding Bass–Serre tree attached to the valuation (Kitayama, 2014). The fundamental valuation–stabilizer equivalence states that

$M$8

(Garden et al., 2024).

From this action one constructs an essential surface associated to the ideal point. In the torus-boundary case, each ideal point produces an essential surface $M$9 in $3$0 (Garden et al., 2024). If some peripheral element has $3$1, then $3$2 has nonempty boundary on $3$3 and its slope $3$4 is strongly detected; if all peripheral traces are bounded at $3$5, the surface detected may be closed, and any boundary slope is at best weakly detected (Garden et al., 2024). The closely related formulation in the knot-manifold setting states that $3$6 can be chosen closed if and only if all peripheral functions are regular at $3$7, while otherwise there is a unique slope $3$8 such that $3$9 is regular at $\partial M$0 and every component of $\partial M$1 represents $\partial M$2 (Chesebro, 2012).

This yields the weighted boundary-slope formula. If $\partial M$3 has boundary slope $\partial M$4 and multiplicity $\partial M$5, then for any slope $\partial M$6,

$\partial M$7

where $\partial M$8 is the algebraic intersection form on $\partial M$9 (Garden et al., 2024). In parallel notation, one also finds

$\pi_1(M)$0

with $\pi_1(M)$1 a positive integer weight and $\pi_1(M)$2 the detected boundary slope (Bénard, 2016). Likewise,

$\pi_1(M)$3

for positive integer weights $\pi_1(M)$4 (Katerba, 2018).

These expressions show that the seminorm measures total boundary width across the essential surfaces detected by ideal points (Garden et al., 2024). A plausible implication is that the piecewise-linearity of the norm is best understood not as an abstract convexity phenomenon, but as the sum of finitely many absolute-value intersection functionals attached to detected surfaces.

4. Equivalent formulations: eigenvalues, polygons, and the $\pi_1(M)$5–polynomial

A third formulation uses peripheral eigenvalues. Let $\pi_1(M)$6 and $\pi_1(M)$7 denote the eigenvalue functions on the eigenvalue variety, giving the eigenvalues of $\pi_1(M)$8 and $\pi_1(M)$9 on the common invariant line. At an ideal point $k$0,

$k$1

and

$k$2

This is especially robust in positive characteristic, including characteristic $k$3 (Garden et al., 2024). The equivalence with the trace and boundary-slope formulations comes from the description of translation lengths in the Bruhat–Tits tree and the abelian nature of the peripheral subgroup (Garden et al., 2024).

In the classical complex case, the eigenvalue variety leads to the $k$4–polynomial. If $k$5 is the $k$6–polynomial and $k$7 its Newton polygon, then the unit ball of the sum of Culler–Shalen seminorms is dual to $k$8: widths in the direction $k$9 are given by the support function

$\mathrm{SL}_2(k)$0

up to normalization and componentwise multiplicities (Garden et al., 2024). In particular, the boundary slopes are precisely the slopes of the edges of $\mathrm{SL}_2(k)$1 (Garden et al., 2024). Equivalent statements appear in the module-theoretic literature: the Newton polygon of the $\mathrm{SL}_2(k)$2–polynomial is dual to the unit ball of the norm, and the seminorm is the support function of the polygon (Chesebro, 2012, Katerba, 2018).

For hyperbolic knot manifolds, the norm on the principal component can also be written as

$\mathrm{SL}_2(k)$3

where $\mathrm{SL}_2(k)$4 is the slope corresponding to $\mathrm{SL}_2(k)$5, the $\mathrm{SL}_2(k)$6 range over all boundary slopes, and the weights $\mathrm{SL}_2(k)$7 are non-negative even integers, with $\mathrm{SL}_2(k)$8 whenever $\mathrm{SL}_2(k)$9 is associated to an ideal point (Ichihara, 14 Jul 2025). In modern language, this norm is equivalent to the support function of a centrally symmetric convex polygon dual to the Newton polygon of the $\mathrm{SL}_2$00–polynomial when that polygon detects both peripheral directions (Ichihara, 14 Jul 2025).

The equivalence of these descriptions is central to computation. One may compute the norm from pole orders on the smooth projective model, from boundary-slope data of essential surfaces, from eigenvalue valuations, or from widths of the Newton polygon of the $\mathrm{SL}_2$01–polynomial (Garden et al., 2024). This suggests that the Culler–Shalen norm is best regarded as a single invariant appearing in several coordinate systems rather than as a construction tied to one preferred formalism.

5. Seminorm versus norm, detection, and module-theoretic interpretations

The function $\mathrm{SL}_2$02 is always a seminorm on $\mathrm{SL}_2$03: it is nonnegative, homogeneous, and satisfies the triangle inequality through convexity of the sum of absolute values (Garden et al., 2024). It fails to be positive definite precisely when some peripheral trace function $\mathrm{SL}_2$04 is constant on $\mathrm{SL}_2$05 (Garden et al., 2024). Geometrically, this happens when all detected surfaces are closed or when all boundary slopes coincide so that the seminorm vanishes along the corresponding line (Garden et al., 2024). In the extreme case, if every $\mathrm{SL}_2$06 is bounded at every ideal point, then $\mathrm{SL}_2$07 detects only closed surfaces and $\mathrm{SL}_2$08 (Garden et al., 2024).

A curve component is called a norm curve if $\mathrm{SL}_2$09 is nonconstant for every nontrivial peripheral element $\mathrm{SL}_2$10 (Chesebro, 2012, Katerba, 2018). Hyperbolic components are norm curves (Chesebro, 2012). For norm curves, the Culler–Shalen seminorm is a true norm on $\mathrm{SL}_2$11, and the unit ball is a genuine centrally symmetric convex polygon whose edges correspond to detected slopes (Chesebro, 2012).

Chesebro’s module-theoretic approach relates this norm to finite generation properties of the coordinate ring over subalgebras generated by a peripheral trace (Chesebro, 2012). When $\mathrm{SL}_2$12 is finite free over $\mathrm{SL}_2$13, the rank equals the degree of the morphism $\mathrm{SL}_2$14, hence

$\mathrm{SL}_2$15

and therefore

$\mathrm{SL}_2$16

whenever $\mathrm{SL}_2$17 is finite free over $\mathrm{SL}_2$18 (Chesebro, 2012). This gives a direct computational path from module ranks to seminorm values (Chesebro, 2012).

A further refinement incorporates the field of definition of the irreducible component. If $\mathrm{SL}_2$19 does not detect a closed essential surface and $\mathrm{SL}_2$20 is not a boundary slope, then for subfields $\mathrm{SL}_2$21 the rank of the trace ring module over $\mathrm{SL}_2$22 is either $\mathrm{SL}_2$23 or $\mathrm{SL}_2$24, depending on the relation between $\mathrm{SL}_2$25 and the minimal field of definition $\mathrm{SL}_2$26 (Katerba, 2018). This result identifies the module ranks with the Culler–Shalen norm up to the field-of-definition factor (Katerba, 2018).

These algebraic criteria also detect geometry. The rank functions are identically infinite if and only if the curve detects a closed essential surface; otherwise, infinite rank at a slope $\mathrm{SL}_2$27 occurs exactly when the curve detects the boundary slope $\mathrm{SL}_2$28 (Chesebro, 2012). A plausible implication is that the Culler–Shalen norm is not merely computable from module structures, but is itself a measure of how far a peripheral trace map is from losing finiteness.

6. Arbitrary characteristic, characteristic $\mathrm{SL}_2$29, and computational examples

The arbitrary-characteristic generalization shows that the constructions needed to define and compute the norm carry over essentially verbatim from the classical complex case (Garden et al., 2024). Let $\mathrm{SL}_2$30 be an algebraically closed field. The valuation theory at ideal points, the action on Bruhat–Tits trees, and the formulas for the seminorm all work over $\mathrm{SL}_2$31 (Garden et al., 2024). In positive characteristic one can define the $\mathrm{SL}_2$32–polynomial $\mathrm{SL}_2$33 and the eigenvalue variety by the same elimination and Gröbner basis approach, and the boundary slopes theorem holds (Garden et al., 2024). For all but finitely many primes $\mathrm{SL}_2$34, the reduced complex $\mathrm{SL}_2$35–polynomial and $\mathrm{SL}_2$36 have the same Newton polygon, hence detect the same strongly detected slopes (Garden et al., 2024).

Characteristic $\mathrm{SL}_2$37 requires special care because trace identities change: parabolics have trace $\mathrm{SL}_2$38, some trace identities collapse, and components may arise on which all single traces vanish identically (Garden et al., 2024). The paper provides characteristic-$\mathrm{SL}_2$39 trace relations and shows that long and short trace coordinates still parametrize the variety of characters, with a birational inverse on each component built from valuations of a nonvanishing trace or a nonvanishing double-trace (Garden et al., 2024). For norm computations, the eigenvalue formulation

$\mathrm{SL}_2$40

is the most robust, because it avoids sign ambiguities and remains well-defined when trace pathologies occur (Garden et al., 2024).

Several examples illustrate the characteristic dependence of detection. For the one-cusped manifold $\mathrm{SL}_2$41, the paper lists explicit $\mathrm{SL}_2$42–polynomials $\mathrm{SL}_2$43, $\mathrm{SL}_2$44, and $\mathrm{SL}_2$45; in characteristic $\mathrm{SL}_2$46, $\mathrm{SL}_2$47 differs from $\mathrm{SL}_2$48 by a factor $\mathrm{SL}_2$49, but the Newton polygons coincide, so the strongly detected boundary slopes and the seminorm’s unit ball are unchanged across characteristics $\mathrm{SL}_2$50, $\mathrm{SL}_2$51, and $\mathrm{SL}_2$52 (Garden et al., 2024). In the same example, the support of $\mathrm{SL}_2$53 contains monomials with exponents

$\mathrm{SL}_2$54

so the width in the meridian direction $\mathrm{SL}_2$55 is $\mathrm{SL}_2$56 and in the longitude direction $\mathrm{SL}_2$57 is $\mathrm{SL}_2$58 (Garden et al., 2024).

The closed hyperbolic manifold $\mathrm{SL}_2$59 has $\mathrm{SL}_2$60 consisting of finitely many points in characteristic $\mathrm{SL}_2$61, and no irreducible characters in characteristic $\mathrm{SL}_2$62; consequently, no ideal points arise on irreducible components, and no essential surfaces are detected in any characteristic by $\mathrm{SL}_2$63–character curves (Garden et al., 2024). By contrast, the closed manifold $\mathrm{SL}_2$64 has $\mathrm{SL}_2$65 containing a curve of irreducible characters, whereas in characteristic $\mathrm{SL}_2$66 it has only finitely many points; the curve yields ideal points and detected essential surfaces only in characteristic $\mathrm{SL}_2$67 (Garden et al., 2024).

Beyond the arbitrary-characteristic setting, several later works situate the norm in broader structures. Twisted Alexander theory shows that ideal points giving Thurston norm–minimizing non-separating surfaces impose regularity constraints on the highest-degree coefficient of the torsion polynomial function, namely

$\mathrm{SL}_2$68

(Kitayama, 2014). The adjoint Reidemeister torsion form on the augmented character variety has vanishing order at ideal points bounded above by the Euler characteristic of the associated essential surface (Bénard, 2016). More recently, "Euclidean lengths and the Culler-Shalen norms of slopes" establishes inequalities between the Euclidean length on a horotorus and the Culler–Shalen norm for certain families of hyperbolic knot exteriors, including

$\mathrm{SL}_2$69

for hyperbolic two-bridge knot exteriors and for exteriors of $\mathrm{SL}_2$70–pretzel knots with $\mathrm{SL}_2$71 odd and $\mathrm{SL}_2$72 (Ichihara, 14 Jul 2025).

Taken together, these results present the Culler–Shalen norm as an invariant at the intersection of valuation theory, character varieties, essential-surface theory, and boundary-slope detection. Algebraically it is the degree of peripheral trace maps; topologically it is the total weighted intersection with boundary slopes of essential surfaces arising from ideal points; and in eigenvalue coordinates it is the sum of absolute values of peripheral valuation functionals (Garden et al., 2024). The arbitrary-characteristic theory shows that this structure persists beyond the classical complex case, even when trace identities and the $\mathrm{SL}_2$73–polynomial change in characteristic $\mathrm{SL}_2$74 (Garden et al., 2024).