Intermediate geodesic growth in virtually nilpotent groups

TL;DR

The paper develops a criterion to classify geodesic growth in virtually s-step nilpotent groups using a polytope construction P(S) derived from Schreier-graph cycles. It shows subexponential growth when no two cycle-axes align on a facet and provides explicit α_s-based bounds, with polynomial behavior in the low-step cases and intermediate growth possible for s≥3. A landmark result is the first known example of intermediate geodesic growth, achieved by a virtually 3-step nilpotent group with γ_geod(n) ≍ exp(n^{3/5} log n). The authors also extend volume-growth asymptotics to virtually nilpotent groups and investigate the Engel group geometry, linking geodesic behavior to broader questions about growth types in nilpotent contexts. The work thus deepens understanding of the geometric structure underlying geodesic growth and frames several open problems on polynomial versus intermediate growth regimes.

Abstract

We give a criterion on pairs $(G,S)$ - where $G$ is a virtually $s$-step nilpotent group and $S$ is a finite generating set - saying whether the geodesic growth is exponential or strictly sub-exponential. Whenever $s=1,2$, this goes further and we prove the geodesic growth is either exponential or polynomial. For $s\ge 3$ however, intermediate growth is possible. We provide an example of virtually $3$-step nilpotent group for which $γ_{\mathrm{geod}}(n) \asymp \exp\!\big(n^{3/5}\cdot \log(n)\big)$. This is the first known example of group with intermediate geodesic growth. Along the way, we prove results on the geometry of virtually nilpotent groups, including asymptotics with error terms for their volume growth.

Figures (8)

• Figure 1: $A=\{(1,0),(0,1),(-1,-1)\}$ in purple and $P(S)$ in green.
• Figure 2: A path decomposed as a product of "freeze frames" of the loop-easure algorithm.
• Figure 3: $x_i=t_i\,a_i\,t_i^{-1}$ and $x_{i+1}=t_{i+1}\,a_{i+1}\,t_{i+1}^{-1}=t_i\,b_{i+1}\,t_i^{-1}$. The limit case $t_i=t_{i+1}$.
• Figure 4: Pictures of $g$, $h$ and $gh$ and some winding numbers.
• Figure 5: A word over $X$ and an equivalent path. Both satisfy $\hat{g}=(24,0)$ and $A(g)=B_y(g)=0$.

Theorems & Definitions (39)

• Theorem : bridson2012groups
• Theorem 1
• Corollary 2
• proof
• Remark 3
• Theorem 4
• Theorem 5: Corollary \ref{['cor:vol']}
• Definition 1.1
• Definition 1.2: $\mathbb R$-words
• Definition 1.3: Stoll metric