Dehn Quasi-Homothetic Functions

• Dehn quasi-homothetic functions are large-scale filling invariants defined via Lipschitz cycles, capturing homotopy and null-homotopy behavior under quasi-isometry.
• They reveal that for horospheres in Hadamard spaces and corresponding lattices, higher-order Dehn functions exhibit Euclidean filling behavior up to a critical dimension.
• Infinite presentations produce fine filling-growth distinctions, demonstrating logarithmic, n log n, and polynomial bounds that yield continuum many asymptotic behaviors.

Dehn quasi-homothetic functions are Dehn-type filling functions viewed up to the coarse equivalences preserved by quasi-isometries and coarse scalings. In the literature considered here, the phrase can be naturally interpreted in two closely related senses: as higher-order or relative Dehn functions stable under quasi-isometry of spaces or pairs, and as fine filling-growth classes arising from infinite presentations, where one studies upper envelopes and lower peaks rather than exact pointwise formulas. Under this interpretation, the subject concerns how null-homotopies, relative fillings, and higher-dimensional fillings behave at large scale, and how those behaviors encode large-scale geometry (Link, 2015, Hughes et al., 2021, Kapovich, 31 May 2026).

1. Definitions and coarse equivalence

For a proper geodesic metric space $X$, higher-order Dehn functions are defined in the Lipschitz filling formalism used by Gromov and Young. If $z$ is a Lipschitz $k$-cycle in $X$, its filling volume is

$$\mathrm{FillVol}_{k+1}(z) =\inf\{\,\mathbf{M}(b)\mid b \text{ a Lipschitz } (k+1)\text{-chain with }\partial b = z\},$$

and the $k$-th order filling function is

$\delta_k^X(V) := \sup\{\mathrm{FillVol}_{k+1}(z)\mid z\ \text{a Lipschitz %%%%5%%%%-cycle in } X,\ \mathbf{M}(z)\le V\}.$

For $k=1$, this is the classical Dehn function; for $k\ge 2$, it is the higher-order filling function. These functions are studied only up to asymptotic equivalence, so if $X$ and $z$0 are quasi-isometric, then $z$1. In that sense, these are “Dehn quasi-isometric” or “Dehn quasi-homothetic” functions: they describe large-scale homotopy geometry in a form stable under quasi-isometries and coarse scalings (Link, 2015).

In the relative setting, for a finitely generated group $z$2 and a collection of subgroups $z$3, the relative Dehn function $z$4 is defined from a finite relative presentation by counting only the $z$5-cells in an Osin–Cayley complex. The standard coarse comparison is $z$6 if there exist constants $z$7 such that

$z$8

and $z$9 means mutual domination. A relative Dehn function is well-defined when it takes only finite values and is determined up to this equivalence across finite relative presentations (Hughes et al., 2021).

For possibly infinite presentations $k$0 with $k$1 finite and $k$2 arbitrary, the Dehn function is still

$k$3

where each defining relator in $k$4 has unit cost, independent of its length. Because the usual coarse comparison collapses all sublinear functions to linear, a finer relation is introduced: $k$5 if

$k$6

and $k$7 in the fine sense if domination holds ორივ directions. This preserves distinctions such as

$k$8

and the paper calls the resulting behavior fine filling-growth (Kapovich, 31 May 2026).

2. Quasi-isometry of pairs and relative filling

A quasi-isometry of pairs

$k$9

is a quasi-isometry of the ambient groups together with coarse control of peripheral cosets: every coset of a subgroup in $X$0 is mapped near a coset of a subgroup in $X$1, and conversely. Within this framework, the central theorem states that if $X$2 is a refinement of $X$3 and $X$4 is well-defined, then $X$5 is well-defined and

$X$6

Relative Dehn functions are therefore quasi-isometry invariants of pairs, up to passing to the canonical refinement $X$7 (Hughes et al., 2021).

The proof proceeds through the geometry of coned-off Cayley graphs. If the peripheral collections are reduced, a quasi-isometry of pairs induces a quasi-isometry of coned-off Cayley graphs

$X$8

and fineness of $X$9 implies fineness of $$\mathrm{FillVol}_{k+1}(z) =\inf\{\,\mathbf{M}(b)\mid b \text{ a Lipschitz } (k+1)\text{-chain with }\partial b = z\},$$0. Almost malnormality is also preserved modulo refinement. These results are then combined with the comparison between relative Dehn functions and coarse isoperimetric functions of coned-off Cayley graphs (Hughes et al., 2021).

A parallel structural theorem concerns cocompact simply connected combinatorial $$\mathrm{FillVol}_{k+1}(z) =\inf\{\,\mathbf{M}(b)\mid b \text{ a Lipschitz } (k+1)\text{-chain with }\partial b = z\},$$1-$$\mathrm{FillVol}_{k+1}(z) =\inf\{\,\mathbf{M}(b)\mid b \text{ a Lipschitz } (k+1)\text{-chain with }\partial b = z\},$$2-complexes with finite edge stabilizers. In that setting, the combinatorial Dehn function $$\mathrm{FillVol}_{k+1}(z) =\inf\{\,\mathbf{M}(b)\mid b \text{ a Lipschitz } (k+1)\text{-chain with }\partial b = z\},$$3 takes only finite values if and only if the $$\mathrm{FillVol}_{k+1}(z) =\inf\{\,\mathbf{M}(b)\mid b \text{ a Lipschitz } (k+1)\text{-chain with }\partial b = z\},$$4-skeleton $$\mathrm{FillVol}_{k+1}(z) =\inf\{\,\mathbf{M}(b)\mid b \text{ a Lipschitz } (k+1)\text{-chain with }\partial b = z\},$$5 is a fine graph. This ties well-defined filling behavior directly to Bowditch fineness, and makes fineness a homotopical criterion rather than merely a graph-theoretic one (Hughes et al., 2021).

The same framework applies to hyperbolically embedded subgroups. If $$\mathrm{FillVol}_{k+1}(z) =\inf\{\,\mathbf{M}(b)\mid b \text{ a Lipschitz } (k+1)\text{-chain with }\partial b = z\},$$6 and $$\mathrm{FillVol}_{k+1}(z) =\inf\{\,\mathbf{M}(b)\mid b \text{ a Lipschitz } (k+1)\text{-chain with }\partial b = z\},$$7 is finitely presented, then the relative Dehn function $$\mathrm{FillVol}_{k+1}(z) =\inf\{\,\mathbf{M}(b)\mid b \text{ a Lipschitz } (k+1)\text{-chain with }\partial b = z\},$$8 is well-defined. By contrast, the Baumslag–Solitar example in the appendix shows that well-definedness is subtler than hyperbolic embedding: for

$$\mathrm{FillVol}_{k+1}(z) =\inf\{\,\mathbf{M}(b)\mid b \text{ a Lipschitz } (k+1)\text{-chain with }\partial b = z\},$$9

the relative Dehn function with respect to $k$0 is well-defined if and only if neither $k$1 divides $k$2 nor $k$3 divides $k$4 (Hughes et al., 2021).

3. Horospheres in products of Hadamard spaces

In the higher-order setting, the ambient space is a product of locally compact Hadamard spaces

$k$5

with Euclidean product metric

$k$6

Boundary points are described by slopes $k$7, where $k$8 and $k$9. A boundary point is regular when all coordinates are positive, and singular when some coordinate is $\delta_k^X(V) := \sup\{\mathrm{FillVol}_{k+1}(z)\mid z\ \text{a Lipschitz %%%%5%%%%-cycle in } X,\ \mathbf{M}(z)\le V\}.$0. If $\delta_k^X(V) := \sup\{\mathrm{FillVol}_{k+1}(z)\mid z\ \text{a Lipschitz %%%%5%%%%-cycle in } X,\ \mathbf{M}(z)\le V\}.$1, the associated Busemann function decomposes as

$\delta_k^X(V) := \sup\{\mathrm{FillVol}_{k+1}(z)\mid z\ \text{a Lipschitz %%%%5%%%%-cycle in } X,\ \mathbf{M}(z)\le V\}.$2

where $\delta_k^X(V) := \sup\{\mathrm{FillVol}_{k+1}(z)\mid z\ \text{a Lipschitz %%%%5%%%%-cycle in } X,\ \mathbf{M}(z)\le V\}.$3 and $\delta_k^X(V) := \sup\{\mathrm{FillVol}_{k+1}(z)\mid z\ \text{a Lipschitz %%%%5%%%%-cycle in } X,\ \mathbf{M}(z)\le V\}.$4. The horosphere centered at $\delta_k^X(V) := \sup\{\mathrm{FillVol}_{k+1}(z)\mid z\ \text{a Lipschitz %%%%5%%%%-cycle in } X,\ \mathbf{M}(z)\le V\}.$5 is then

$\delta_k^X(V) := \sup\{\mathrm{FillVol}_{k+1}(z)\mid z\ \text{a Lipschitz %%%%5%%%%-cycle in } X,\ \mathbf{M}(z)\le V\}.$6

The integer $\delta_k^X(V) := \sup\{\mathrm{FillVol}_{k+1}(z)\mid z\ \text{a Lipschitz %%%%5%%%%-cycle in } X,\ \mathbf{M}(z)\le V\}.$7 records the number of factors in which the defining geodesic ray projects nontrivially (Link, 2015).

The key geometric construction is the family of slices inside $\delta_k^X(V) := \sup\{\mathrm{FillVol}_{k+1}(z)\mid z\ \text{a Lipschitz %%%%5%%%%-cycle in } X,\ \mathbf{M}(z)\le V\}.$8. A $\delta_k^X(V) := \sup\{\mathrm{FillVol}_{k+1}(z)\mid z\ \text{a Lipschitz %%%%5%%%%-cycle in } X,\ \mathbf{M}(z)\le V\}.$9-slice has the form

$k=1$0

where exactly $k=1$1 of the factors are whole spaces $k=1$2, the remaining active factors are geodesic lines asymptotic to $k=1$3, and inactive factors are fixed at the basepoint. The main technical proposition states that each $k=1$4-slice is bilipschitz equivalent to

$k=1$5

with bilipschitz constant depending only on $k=1$6. In particular, every $k=1$7-slice is bilipschitz to $k=1$8 (Link, 2015).

This bilipschitz control is precisely what Young’s filling construction requires. Using Young’s simplicial extension criterion, the paper proves that every horosphere centered at $k=1$9 is Lipschitz $k\ge 2$0-connected. Consequently, the horosphere is undistorted up to dimension $k\ge 2$1, and for each $k\ge 2$2,

$k\ge 2$3

These estimates are sharp: Euclidean lower bounds are inherited from the Euclidean factors present in the slices and ambient product (Link, 2015).

When $k\ge 2$4 is regular, one has $k\ge 2$5. The corresponding statement is the paper’s Theorem A: any horosphere centered at a regular boundary point of a product of $k\ge 2$6 locally compact Hadamard spaces is undistorted up to dimension $k\ge 2$7, and for any $k\ge 2$8, the $k\ge 2$9-th order Dehn function is asymptotic to $X$0 (Link, 2015).

4. Lattices, cusps, and rational rank one

The horospherical results transfer from spaces to groups by quasi-isometry. Let $X$1 be a lattice acting cocompactly on

$X$2

where the $X$3 are disjoint horoballs. If all horoball centers are represented by geodesic rays that do not project to a point in at least $X$4 factors, then $X$5 is undistorted up to dimension $X$6, and for each $X$7,

$X$8

The passage from horospheres to $X$9 uses the quasi-isometry invariance of higher-order Dehn functions and undistortedness (Link, 2015).

An important special case is that of irreducible $z$00-rank one lattices acting on a product $z$01 of symmetric spaces of noncompact type. By the description attributed to Prasad, Raghunathan, and Leuzinger–Pittet, such a lattice acts cocompactly on the complement of finitely many disjoint horoballs, and the horoball centers lie in the regular boundary. Hence $z$02, and the resulting theorem states that $z$03 is undistorted up to dimension $z$04, while for every $z$05,

$z$06

The paper notes that this covers all irreducible $z$07-rank one lattices acting on such products, with Hilbert modular groups having already been treated by Young (Link, 2015).

In this setting, quasi-homothetic language is especially natural. The large-scale filling behavior of the group matches that of Euclidean space up to a rank threshold determined by the number of active factors. The boundary slope, via the distinction between regular and singular boundary points, directly controls the dimension up to which Euclidean filling persists (Link, 2015).

5. Infinite presentations and fine filling-growth

Infinite presentations introduce a different, but related, notion of quasi-homothetic behavior. For

$z$08

with $z$09 finite and $z$10 possibly infinite, each relator has unit cost regardless of length. This changes the rigidity of Dehn functions dramatically. The paper proves that the infinite presentation

$z$11

of $z$12 has Dehn function of order $z$13: $z$14 for all sufficiently large $z$15. The upper bound comes from a dyadic strip decomposition of lattice polygons; the lower bound comes from a signed binary weight $z$16 controlling how many signed powers of $z$17 are needed in each horizontal row (Kapovich, 31 May 2026).

The general construction uses a scalar relation invariant

$z$18

where $z$19, assumed to be a conjugacy-invariant homomorphism with infinite image and polynomial growth

$z$20

Choosing a cyclically reduced relation $z$21 with $z$22, one obtains a logarithmic presentation by adjoining all cyclically reduced relators in $z$23 together with dyadic powers $z$24. The corresponding Dehn function satisfies

$z$25

in the fine sense (Kapovich, 31 May 2026).

For every $z$26, the construction yields an infinite presentation

$z$27

with

$z$28

for all $z$29, and, provided the hard values $z$30 are $z$31-efficiently realized by $z$32, there exists an infinite sequence $z$33 such that

$z$34

For $z$35, signed area gives $z$36, so every exponent $z$37 occurs. For closed orientable surface groups, the fundamental-class invariant $z$38 gives $z$39, so every exponent $z$40 occurs (Kapovich, 31 May 2026).

These examples show that, in contrast with the finite-presentation setting, infinite presentations of a fixed finitely generated group on a fixed generating set can exhibit continuum many distinct fine filling-growth behaviors. They also show that quasi-homothetic behavior need not mean a single exact asymptotic law at all scales: the relevant pattern can instead be a global envelope together with matching lower-bound peaks along an infinite subsequence. At the same time, the paper emphasizes that not every infinite relator family produces intermediate growth: the presentation

$z$41

has linear Dehn function (Kapovich, 31 May 2026).

6. Interpretive scope and recurring themes

Across these three frameworks, the phrase “Dehn quasi-homothetic functions” denotes filling invariants that are meaningful only up to coarse comparison. In products of Hadamard spaces and their lattices, the quasi-homothetic class is Euclidean: $z$42 up to a dimension determined by the number of active factors in the boundary slope. For pairs $z$43, the relative Dehn function is preserved under quasi-isometries of pairs after passage to a refinement $z$44. For infinite presentations, fine filling-growth replaces coarse quasi-isometry invariance by a finer asymptotic comparison that retains logarithmic and sublinear distinctions (Link, 2015, Hughes et al., 2021, Kapovich, 31 May 2026).

Several recurrent misconceptions are excluded by these results. First, quasi-homothetic does not mean exact self-similarity or exact power-law scaling at every length scale: the infinite-presentation constructions produce upper envelopes and matching peaks, not uniform pointwise equivalence at all $z$45 (Kapovich, 31 May 2026). Second, it does not mean presentation-independence in arbitrary generality: for a fixed group and fixed generating set, distinct infinite relator sets can realize continuum many distinct fine filling-growth types (Kapovich, 31 May 2026). Third, Euclidean filling behavior is not dimension-free: in the horosphere results, the threshold is $z$46, where $z$47 is the number of factors in which the boundary ray is nonconstant (Link, 2015). Fourth, quasi-isometry invariance in the relative setting depends on the peripheral structure, and the correct invariant is attached to the pair, possibly after refinement, rather than to the ambient group alone (Hughes et al., 2021).

Taken together, these results make the subject a study of large-scale homotopy geometry through filling functions. The unifying principle is that Dehn-type functions become quasi-homothetic precisely when one passes from exact presentations or exact metrics to their coarse asymptotic classes. In one direction this yields rigidity, such as Euclidean higher-order filling for regular horospheres and $z$48-rank one lattices; in another it reveals flexibility, such as logarithmic, $z$49, and $z$50-type fine filling-growth for infinite presentations of classical groups (Link, 2015, Hughes et al., 2021, Kapovich, 31 May 2026).