So I am looking for examples of the following phenomenon.

Suppose that $ V$ is a variety with a computable equational theory which is not locally finite. Suppose that $ G$ is an infinite finitely presented group generated by elements $ s_{1},\dots,s_{k}$ . Suppose furthermore that $ G$ has a polynomial time computable normal form. In other words, there is a function $ f:\{w_{1},\dots,w_{k},w^{-1}_{1},\dots,w_{k}^{-1}\}^{*}\rightarrow\mathbb{N}$ computable in polynomial time such that if $ d_{i},e_{i}\in\{-1,1\},a_{i},b_{i}\in\{1,\dots,k\}$ for each $ i$ , then $ f(w_{a_{1}}^{d_{1}}\dots w_{a_{u}}^{d_{u}})=f(w_{b_{1}}^{e_{1}}\dots w_{b_{v}}^{e_{v}})$ if and only if $ s_{a_{1}}^{d_{1}}\dots s_{a_{u}}^{d_{u}}=s_{b_{1}}^{e_{1}}\dots s_{b_{v}}^{e_{v}}$ . Suppose that there is some $ n$ along with two matrices of terms $ $ (t_{i,j}(x_{1},…,x_{n}))_{1\leq i\leq n,1\leq j\leq k},(t^{*}_{i,j}(x_{1},…,x_{n}))_{1\leq i\leq n,1\leq j\leq k}.$ $ These terms come from the variety $ V$ . Then whenever $ X\in V,$ define actions of $ \{w_{1},\dots,w_{k},w^{-1}_{1},\dots,w_{k}^{-1}\}^{*}$ on $ X^{n}$ $ $ (x_{1},\dots,x_{n})\cdot w_{j}=(t_{1,j}(x_{1},\dots,x_{n}),\dots,t_{n,j}(x_{1},\dots,x_{n}))$ $ and $ $ (x_{1},\dots,x_{n})\cdot w_{j}^{-1}=(t_{1,j}^{*}(x_{1},\dots,x_{n}),\dots,t_{n,j}^{*}(x_{1},\dots,x_{n})).$ $

Then the variety $ V$ satisfies the identities $ $ (x_{1},\dots,x_{n})\cdot w_{a_{1}}^{d_{1}}\dots w_{a_{u}}^{d_{u}}=(x_{1},\dots,x_{n})\cdot w_{b_{1}}^{e_{1}}\dots w_{b_{v}}^{e_{v}}$ $ if and only if $ s_{a_{1}}^{d_{1}}\dots s_{a_{u}}^{d_{u}}=s_{b_{1}}^{e_{1}}\dots s_{b_{v}}^{e_{v}}$ , and the equational theory of the variety $ V$ is axiomatized by the identities of the form $ $ (x_{1},\dots,x_{n})\cdot w_{a_{1}}^{d_{1}}\dots w_{a_{u}}^{d_{u}}=(x_{1},\dots,x_{n})\cdot w_{b_{1}}^{e_{1}}\dots w_{b_{v}}^{e_{v}}$ $ where $ d_{i},e_{i}\in\{-1,1\},a_{i},b_{i}\in\{1,\dots,k\}$ for each $ i.$

In this case, the group $ G$ acts on $ X^{n}$ for each $ X\in V$ by letting $ $ (x_{1},…,x_{n})\cdot s_{a_{1}}^{d_{1}}\dots s_{a_{u}}^{d_{u}}=(x_{1},…,x_{n})\cdot w_{a_{1}}^{d_{1}}\dots w_{a_{u}}^{d_{u}}$ $ where $ d_{i},e_{i}\in\{-1,1\},a_{i},b_{i}\in\{1,\dots,k\}$ for each $ i.$

**Examples**

Example 1: The symmetry group $ S_{k}$ . Suppose that $ \tau_{i}=(i,i+1)$ . Then we set $ $ (x_{1},\dots,x_{k})\tau_{i}=(x_{1},\dots,x_{i-1},x_{i+1},x_{i},x_{i+2},\dots,x_{k}).$ $

Example 2: The braid group $ B_{k}$ acting on racks and quandles. Suppose that $ (X,*)$ is a rack or a quandle (Let us use left-distributivity $ x*(y*z)=(x*y)*(x*z)$ ). Then define $ $ (x_{1},\dots,x_{n})\cdot\sigma_{i}=(x_{1},\dots,x_{i-1},x_{i}*x_{i+1},x_{i},x_{i+2},\dots,x_{n}).$ $

Example 3: $ \mathrm{Aut}(F_{k})$ acting on tuples from groups.

Example 4: Finite direct products of groups.

In the case where the group $ G$ is finite or though the conditions I have listed out are otherwise not technically satisfied (for example by the infinite strand braid group $ B_{\infty}$ on $ X^{\mathbb{N}}$ for each rack $ X$ ), feel free to give an answer anyways if you feel the answer is still interesting yet does not stray from the spirit of the question too much.