$l_2$ and $\mathcal B(l_2)$ be the Banach algebra of bounded linear operators on $l_2$. For every $n\in{\mathbb N}$ let $G_n$ be the subgroup of $\mathcal B(l_2)$ consisting of invertible linear operators $T:l_2\to l_2$ such that

-   $Te_0\in (0,+\infty)\cdot e_0$;

-   $Te_i\in e_i+{\mathbb R}\cdot e_0$ for all $1\le i\le n$;

-   $Te_i=e_i$ for all $i>n$.

The tower $(G_n)_{n\in{\mathbb N}}$ satisfies ${\mathrm{PTA}}$ because each group $G_n$ is locally compact, see [@TSH], [@HSTH]. On the other hand, for every $n\in{\mathbb N}$ the triple $(G_n,G_{n+1},G_{n+2})$ is not bi-balanced. The reason is that for the neighborhood $W=\{T\in G_{n+2}:Te_{n+2}\in e_{n+2}+(-1,1)e_0\}$ of the identity ${\mathrm{id}}$ in $G_{n+2}$ the set $Z=\sqrt[G_n]{W}$ lies in the subgroup $G_{n+1}$. Then for each neighborhood $V\subset G_{n+1}$, the product $ZVZ\subset G_{n+1}$ fails to be a neighborhood of ${\mathrm{id}}$ in the group $G_{n+2}$.

It is clear that each balanced triple of groups is bi-balanced. The converse implication is not true.

In the group $G=GL(3,{\mathbb R})$ of non-degenerated $3\times 3$-matrices consider the subgroups $$\Gamma=\left\{\left(\begin{array}{ccc}
a_{11}&a_{12}&0\\
a_{21}&a_{22}&0\\
0&0&a_{33}\end{array}\right)\in G\right\}\mbox{ \ \ and \ \ }
H=\left\{\left(\begin{array}{ccc}
a_{11}&0&0\\
0&1&0\\
0&0&1\end{array}\right)\in G\right\}.$$ It is easy to check that the triple $(G,\Gamma,H)$ is bi-balanced but not balanced.

Direct limits in the category of uniform spaces {#s:ulim}
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In this 