--- title: Dehn Quasi-Homothetic Functions url: https://www.emergentmind.com/topics/dehn-quasi-homothetic-functions type: topic --- # Dehn Quasi-Homothetic Functions 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 [1508.02192][2107.03355][2606.01290]. ## 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 \(k\)-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 \(Y\) are quasi-isometric, then \(\delta_k^X \asymp \delta_k^Y\). 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 [1508.02192]. In the relative setting, for a finitely generated group \(G\) and a collection of subgroups \(\mathcal P\), the relative Dehn function \(\Delta_{G,\mathcal P}\) is defined from a finite relative presentation by counting only the \(R\)-cells in an Osin–Cayley complex. The standard coarse comparison is \(f\preceq g\) if there exist constants \(C,K,L\in\mathbb N\) such that \[ f(n) \le C\,g(Kn) + L\,n\quad\forall\,n, \] and \(f\asymp g\) 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 [2107.03355]. For possibly infinite presentations \(\mathcal P=\langle X\mid R\rangle\) with \(X\) finite and \(R\) arbitrary, the Dehn function is still \[ \delta_{\mathcal P}(n) =\max\{\operatorname{Area}_{\mathcal P}(w)\mid w=_G1,\ w \text{ freely reduced},\ |w|_X\le n\}, \] where each defining relator in \(R\) has unit cost, independent of its length. Because the usual coarse comparison collapses all sublinear functions to linear, a finer relation is introduced: \(f\precsim g\) if \[ f(n)\le C g(Cn+C)+C\quad\forall n, \] and \(f\asymp g\) in the fine sense if domination holds ორივ directions. This preserves distinctions such as \[ 1 \prec \log n \prec n^\alpha \prec n \prec n\log n \prec n^2,\quad 0<\alpha<1, \] and the paper calls the resulting behavior fine filling-growth [2606.01290]. ## 2. Quasi-isometry of pairs and relative filling A quasi-isometry of pairs \[ q:(G,\mathcal P)\to(H,\mathcal Q) \] is a quasi-isometry of the ambient groups together with coarse control of peripheral cosets: every coset of a subgroup in \(\mathcal P\) is mapped near a coset of a subgroup in \(\mathcal Q\), and conversely. Within this framework, the central theorem states that if \(\mathcal P^*\) is a refinement of \(\mathcal P\) and \(\Delta_{H,\mathcal Q}\) is well-defined, then \(\Delta_{G,\mathcal P^*}\) is well-defined and \[ \Delta_{G,\mathcal P^*} \asymp \Delta_{H,\mathcal Q}. \] Relative Dehn functions are therefore quasi-isometry invariants of pairs, up to passing to the canonical refinement \(\mathcal P^*\) [2107.03355]. 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 \[ \hat q:\hat\Gamma(G,\mathcal P)\to \hat\Gamma(H,\mathcal Q), \] and fineness of \(\hat\Gamma(H,\mathcal Q)\) implies fineness of \(\hat\Gamma(G,\mathcal P)\). 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 [2107.03355]. A parallel structural theorem concerns cocompact simply connected combinatorial \(G\)-\(2\)-complexes with finite edge stabilizers. In that setting, the combinatorial Dehn function \(\Delta_X\) takes only finite values if and only if the \(1\)-skeleton \(X^{(1)}\) 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 [2107.03355]. The same framework applies to hyperbolically embedded subgroups. If \(H\hookrightarrow_h G\) and \(G\) is finitely presented, then the relative Dehn function \(\Delta_{G,H}\) is well-defined. By contrast, the Baumslag–Solitar example in the appendix shows that well-definedness is subtler than hyperbolic embedding: for \[ \mathrm{BS}(k,l)=\langle a,t \mid t a^k t^{-1} = a^l\rangle, \] the relative Dehn function with respect to \(\langle t\rangle\) is well-defined if and only if neither \(k\) divides \(l\) nor \(l\) divides \(k\) [2107.03355]. ## 3. Horospheres in products of Hadamard spaces In the higher-order setting, the ambient space is a product of locally compact Hadamard spaces \[ X = X_1 \times X_2 \times \cdots \times X_r \] with Euclidean product metric \[ d(x,y) = \sqrt{ \sum_{i=1}^r d_i(x_i,y_i)^2 }. \] Boundary points are described by slopes \(\theta=(\theta_1,\dots,\theta_r)\in E\), where \(\theta_i\ge 0\) and \(\|\theta\|=1\). A boundary point is regular when all coordinates are positive, and singular when some coordinate is \(0\). If \(\tilde\xi\in\partial X_\theta\), the associated Busemann function decomposes as \[ b_{\tilde\xi}(x,y) = \sum_{i\in I^+(\theta)} \theta_i\, b_{\xi_i}(x_i,y_i), \] where \(I^+(\theta)=\{i:\theta_i>0\}\) and \(q=|I^+(\theta)|\). The horosphere centered at \(\tilde\xi\) is then \[ \mathcal{H} = \left\{y=(y_1,\dots,y_r)\in X:\ \sum_{i\in I^+(\theta)} \theta_i\, b_{\xi_i}(o_i,y_i)=0\right\}. \] The integer \(q\) records the number of factors in which the defining geodesic ray projects nontrivially [1508.02192]. The key geometric construction is the family of slices inside \(\mathcal H\). A \(k\)-slice has the form \[ S = (s_1\times\cdots\times s_r)\cap \mathcal{H}, \] where exactly \(k\) of the factors are whole spaces \(X_i\), the remaining active factors are geodesic lines asymptotic to \(\xi_i\), and inactive factors are fixed at the basepoint. The main technical proposition states that each \(k\)-slice is bilipschitz equivalent to \[ \prod_{i\in I(S)} X_i \times \mathbb{R}^{\,q-k-1}, \] with bilipschitz constant depending only on \(\theta\). In particular, every \(0\)-slice is bilipschitz to \(\mathbb{R}^{q-1}\) [1508.02192]. 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 \(\tilde\xi\in\partial X_\theta\) is Lipschitz \((q-2)\)-connected. Consequently, the horosphere is undistorted up to dimension \(q-1\), and for each \(k\le q-2\), \[ \delta_k^{\mathcal H}(V) \asymp V^{(k+1)/k}. \] These estimates are sharp: Euclidean lower bounds are inherited from the Euclidean factors present in the slices and ambient product [1508.02192]. When \(\tilde\xi\) is regular, one has \(q=r\). The corresponding statement is the paper’s Theorem A: any horosphere centered at a regular boundary point of a product of \(r\) locally compact Hadamard spaces is undistorted up to dimension \(r-1\), and for any \(k\le r-2\), the \(k\)-th order Dehn function is asymptotic to \(V^{(k+1)/k}\) [1508.02192]. ## 4. Lattices, cusps, and rational rank one The horospherical results transfer from spaces to groups by quasi-isometry. Let \(\Gamma\subset\mathrm{Isom}(X)\) be a lattice acting cocompactly on \[ X\setminus\bigcup_j \mathcal B_j, \] where the \(\mathcal B_j\) are disjoint horoballs. If all horoball centers are represented by geodesic rays that do not project to a point in at least \(q\) factors, then \(\Gamma\) is undistorted up to dimension \(q-1\), and for each \(k\le q-2\), \[ \delta_k^\Gamma(V) \asymp V^{(k+1)/k}. \] The passage from horospheres to \(\Gamma\) uses the quasi-isometry invariance of higher-order Dehn functions and undistortedness [1508.02192]. An important special case is that of irreducible \(\mathbb Q\)-rank one lattices acting on a product \(X\) 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 \(q=r\), and the resulting theorem states that \(\Gamma\) is undistorted up to dimension \(r-1\), while for every \(k\le r-2\), \[ \delta_k^\Gamma(V) \asymp V^{(k+1)/k}. \] The paper notes that this covers all irreducible \(\mathbb Q\)-rank one lattices acting on such products, with Hilbert modular groups having already been treated by Young [1508.02192]. 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 [1508.02192]. ## 5. Infinite presentations and fine filling-growth Infinite presentations introduce a different, but related, notion of quasi-homothetic behavior. For \[ \mathcal P=\langle X\mid R\rangle \] with \(X\) finite and \(R\) 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 \[ \left\langle a,b \ \middle|\ [a^{2^k},b],\ k=0,1,2,\ldots\right\rangle \] of \(\mathbb Z^2\) has Dehn function of order \(n\log n\): \[ c n\log n\le \delta_{\mathcal P}(n)\le C n\log n \] for all sufficiently large \(n\). The upper bound comes from a dyadic strip decomposition of lattice polygons; the lower bound comes from a signed binary weight \(\tau(n)\) controlling how many signed powers of \(2\) are needed in each horizontal row [2606.01290]. The general construction uses a scalar relation invariant \[ \chi:N\to\mathbb Z, \] where \(N=\ker(F(X)\to G)\), assumed to be a conjugacy-invariant homomorphism with infinite image and polynomial growth \[ \Phi_\chi(n)=\max\{|\chi(w)|\mid w\in N,\ |w|_X\le n\}=O(n^D). \] Choosing a cyclically reduced relation \(q\in N\) with \(\chi(q)=1\), one obtains a logarithmic presentation by adjoining all cyclically reduced relators in \(\ker\chi\) together with dyadic powers \(q^{2^k}\). The corresponding Dehn function satisfies \[ \delta_{\mathcal P_{\log}}(n)\asymp \log n \] in the fine sense [2606.01290]. For every \(0<\alpha