I'm curious about non-shperical regions in the Cayley complex of a hyperbolic group $G = < X : R \>$ that one might reasonably consider "curved", but I haven't found much discussion of local curvature in a Cayley complex.

We could for-example look for curvature in the sense of the presentation's isoparametric inequality by asking what bubble-like regions determine the constant as follows.

We regard $\pi_2$ as a $\def\Z{\mathbb{Z}}\Z[G]$-submodule of $\Z[G]^R$ in the usual way. Let $|\cdot|$ denote the $l^1$-norm on $\Z[G]^R$ or $\Z[G]^X$. So $|\cdot|$ assigns an area to an element in $\Z[G]^R$ or a length to an element of $\Z[G]^X$. Also, let $\partial$ denote the Fox derivative sending elements of $\Z[G]^R$ to their boundary in $\Z[G]^X$. So $\pi_2 = \textrm{ker}\ \partial$.

Let $C_k$ denote the set of diagrams in $\Z[G]^R$ such that the minimum area obtainable by adding an element of $\pi_2$ remains larger than $k$ times its boundary length. $$ C_k = \{ x \in \Z[G]^R : |x+y| > k |\partial x| \ \ \forall x \in \pi_2 \} $$ $C_k$ is not a group under the module addition, but it's closed under both multiplication by $\Z[G]$, aka translation, as well as addition with elements of $\pi_2$. There are of course many elements of $C_k$ whose sum remains inside $C_k$ though; in particular $\pi_2$ is contained inside $C_k$ for all $k$.

I'm curious whether these $C_k$ are finitely generated with respect to the module addition and multiplication by $\Z[G]$, as well as whether $C_k = \pi_2$ for $k > n$ for some $n$. It's worth noting that finite generation up to $\Z[G]$ holds for both $C_\infty = \pi_2$ (previously) and $C_m$ where $m$ is the reciprocal of the largest relator length are both finitely generated.

I'm curious about non-shperical regions in the Cayley complex of a hyperbolic group $G = < X : R \>$ that one might reasonably consider "curved", but I haven't found much discussion of local curvature in a Cayley complex.

We could for-example look for curvature in the sense of the presentation's isoparametric inequality by asking what bubble-like regions determine the constant as follows.

We regard $\pi_2$ as a $\def\Z{\mathbb{Z}}\Z[G]$-submodule of $\Z[G]^R$ in the usual way. Let $|\cdot|$ denote the $l^1$-norm on $\Z[G]^R$ or $\Z[G]^X$. So $|\cdot|$ assigns an area to an element in $\Z[G]^R$ or a length to an element of $\Z[G]^X$. Also, let $\partial$ denote the Fox derivative sending elements of $\Z[G]^R$ to their boundary in $\Z[G]^X$. So $\pi_2 = \textrm{ker}\ \partial$.

Let $C_k$ denote the set of diagrams in $\Z[G]^R$ such that the minimum area obtainable by adding an element of $\pi_2$ remains larger than $k$ times its boundary length. $$ C_k = \{ x \in \Z[G]^R : |x+y| > k |\partial x| \ \ \forall x \in \pi_2 \} $$ $C_k$ is not a group under the module addition, but it's closed under both multiplication by $\Z[G]$, aka translation, as well as addition with elements of $\pi_2$. There are of course many elements of $C_k$ whose sum remains inside $C_k$ though; in particular $\pi_2$ is contained inside $C_k$ for all $k$.

I'm curious whether these $C_k$ are finitely generated with respect to the module addition and multiplication by $\Z[G]$, as well as whether $C_k = \pi_2$ for $k > n$ for some $n$. It's worth noting that finite generation up to $\Z[G]$ holds for both $C_\infty = \pi_2$ (previously) and $C_m$ where $m$ is the reciprocal of the largest relator length are both finitely generated.