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There are some nice examples in knot theory and quantum algebra.

If $X$ is an algebraic variety over $\mathbb{Q}$, there is a canonical outer action $G_{\mathbb{Q}} \rightarrow Out(\hat{\pi}_1(X(\mathbb{C})))$ (which also exists for other fields than $\mathbb{Q}$ using the so-called algebraic fundamental group but I don't want to say something wrong about that). Roughly speaking, this is because finite covering of $X$ can be defined over $\bar{\mathbb{Q}}$, together with the relation between (regular) finite covering and finite quotients of the fundamental group. This action thus has the same origine as for dessin d'enfants.

Of particular interest is the study of this action in the case $X$ is the moduli space of algebraic curves of genus $g$ with $n$ marked points. This was suggested in Grothendieck's esquisse, and leads to the so-called Grothendieck-Teichmuller theory which gives a rather explicit description of a group which actually contains $G_{\mathbb{Q}}$. (see Cartographic group and flat stringy connection or Where is a good place to start learning about the Grothendieck-Teichmuller group?)

Actually, there are several flavours of the Grothendieck-Teichmuller group: a profinite one $\widehat{GT}$ which does contain $G_{\mathbb{Q}}$ and a group $GT(k)$ defined for every field $k$. A deep result of Drinfeld assert that this latter group is in some sense a universal automorphism group of braided monoidal categories. Indeed, it acts on the set of Drinfeld associator with coefficients in $k$.

Now, there is also a morphism

$G_{\mathbb{Q}}\rightarrow GT(\mathbb{Q}_{\ell})$

for every prime number $\ell$. Hence the absolute galois group acts on each kind of object in which associators come up (assuming that it is in a situation where one can work over $\mathbb{Q}_{\ell}$). It leads to, I think, quite surprizing examples like action on finite type invariants of knots and links, on quantization functor of Lie bialgebras, and several other constructions arising in deformation/quantization theory as they are often related to Drinfeld associators.

There are some nice examples in knot theory and quantum algebra.

If $X$ is an algebraic variety over $\mathbb{Q}$, there is a canonical outer action $G_{\mathbb{Q}} \rightarrow Out(\hat{\pi}_1(X(\mathbb{C})))$ (which also exists for other fields than $\mathbb{Q}$ using the so-called algebraic fundamental group but I don't want to say something wrong about that). Roughly speaking, this is because finite covering of $X$ can be defined over $\bar{\mathbb{Q}}$, together with the relation between (regular) finite covering and finite quotients of the fundamental group. This action thus has the same origine as for dessin d'enfants.

Of particular interest is the study of this action in the case $X$ is the moduli space of algebraic curves of genus $g$ with $n$ marked points. This was suggested in Grothendieck's esquisse, and leads to the so-called Grothendieck-Teichmuller theory which gives a rather explicit description of a group which actually contains $G_{\mathbb{Q}}$. (see Cartographic group and flat stringy connection or Where is a good place to start learning about the Grothendieck-Teichmuller group?)

Actually, there are several flavours of the Grothendieck-Teichmuller group: a profinite one $\widehat{GT}$ which does contain $G_{\mathbb{Q}}$ and a group $GT(k)$ defined for every field $k$. A deep result of Drinfeld assert that this latter group is in some sense a universal automorphism group of braided monoidal categories. Indeed, it acts on the set of Drinfeld associator with coefficients in $k$.

Now, there is also a morphism

$G_{\mathbb{Q}}\rightarrow GT(\mathbb{Q}_{\ell})$

for every prime number $\ell$. Hence the absolute galois group acts on each kind of object in which associators come up (assuming that it is in a situation where one can work over $\mathbb{Q}_{\ell}$). It leads to, I think, quite surprizing examples like action on finite type invariants of knots and links, on quantization functor of Lie bialgebras, and several other constructions arising in deformation/quantization theory as they are often related to Drinfeld associators.

There are some nice examples in knot theory and quantum algebra.

If $X$ is an algebraic variety over $\mathbb{Q}$, there is a canonical outer action $G_{\mathbb{Q}} \rightarrow Out(\hat{\pi}_1(X(\mathbb{C})))$ (which also exists for other fields than $\mathbb{Q}$ using the so-called algebraic fundamental group but I don't want to say something wrong about that). Roughly speaking, this is because finite covering of $X$ can be defined over $\bar{\mathbb{Q}}$, together with the relation between (regular) finite covering and finite quotients of the fundamental group. This action thus has the same origine as for dessin d'enfants.

Or particular interest is the study of this action in the case $X$ is the moduli space of algebraic curves of genus $g$ with $n$ marked points. This was suggested in Grothendieck's esquisse, and leads to the so-called Grothendieck-Teichmuller theory which gives a rather explicit description of a group which actually contains $G_{\mathbb{Q}}$. (see Cartographic group and flat stringy connection or Where is a good place to start learning about the Grothendieck-Teichmuller group?)

Actually, there are several flavours of the Grothendieck-Teichmuller group: a profinite one $\widehat{GT}$ which does contain $G_{\mathbb{Q}}$ and a group $GT(k)$ defined for every field $k$. A deep result of Drinfeld assert that this latter group is in some sense a universal automorphism group of braided monoidal categories. Indeed, it acts on the set of Drinfeld associator with coefficients in $k$.

Now, there is also a morphism

$G_{\mathbb{Q}}\rightarrow GT(\mathbb{Q}_{\ell})$

for every prime numbre $\ell$. Hence the absolute galois group acts on each kind of object in which associators come up (assuming that it is in a situation where one can work over $\mathbb{Q}_{\ell}$). It leads to, I think, quite surprizing examples like action on finite type invariants of knots and links, on quantization functor of Lie bialgebras, and several other constructions arising in deformation/quantization theory as they are often related to Drinfeld associators.

There are some nice examples in knot theory and quantum algebra.

If $X$ is an algebraic variety over $\mathbb{Q}$, there is a canonical outer action $G_{\mathbb{Q}} \rightarrow Out(\hat{\pi}_1(X(\mathbb{C})))$ (which also exists for other fields than $\mathbb{Q}$ using the so-called algebraic fundamental group but I don't want to say something wrong about that). Roughly speaking, this is because finite covering of $X$ can be defined over $\bar{\mathbb{Q}}$, together with the relation between (regular) finite covering and finite quotients of the fundamental group. This action thus has the same origine as for dessin d'enfants.

Of particular interest is the study of this action in the case $X$ is the moduli space of algebraic curves of genus $g$ with $n$ marked points. This was suggested in Grothendieck's esquisse, and leads to the so-called Grothendieck-Teichmuller theory which gives a rather explicit description of a group which actually contains $G_{\mathbb{Q}}$. (see Cartographic group and flat stringy connection or Where is a good place to start learning about the Grothendieck-Teichmuller group?)

Actually, there are several flavours of the Grothendieck-Teichmuller group: a profinite one $\widehat{GT}$ which does contain $G_{\mathbb{Q}}$ and a group $GT(k)$ defined for every field $k$. A deep result of Drinfeld assert that this latter group is in some sense a universal automorphism group of braided monoidal categories. Indeed, it acts on the set of Drinfeld associator with coefficients in $k$.

Now, there is also a morphism

$G_{\mathbb{Q}}\rightarrow GT(\mathbb{Q}_{\ell})$

for every prime number $\ell$. Hence the absolute galois group acts on each kind of object in which associators come up (assuming that it is in a situation where one can work over $\mathbb{Q}_{\ell}$). It leads to, I think, quite surprizing examples like action on finite type invariants of knots and links, on quantization functor of Lie bialgebras, and several other constructions arising in deformation/quantization theory as they are often related to Drinfeld associators.

If $X$ is an algebraic variety over $\mathbb{Q}$, there is a canonical outer action $G_{\mathbb{Q}} \rightarrow Out(\hat{\pi}_1(X(\mathbb{C})))$ (which also exists for other fields than $\mathbb{Q}$ using the so-called algebraic fundamental group but I don't want to say something wrong about that). Roughly speaking, this is because finite covering of $X$ can be defined over $\bar{\mathbb{Q}}$, together with the relation between (regular) finite covering and finite quotients of the fundamental group. This action thus has the same origine as for dessin d'enfants.

Or particular interest is the study of this action in the case $X$ is the moduli space of algebraic curves of genus $g$ with $n$ marked points. This was suggested in Grothendieck's esquisse, and leads to the so-called Grothendieck-Teichmuller theory which gives a rather explicit description of a group which actually contains $G_{\mathbb{Q}}$. (see Cartographic group and flat stringy connection or Cartographic group and flat stringy connection)

Actually, there are several flavours of the Grothendieck-Teichmuller group: a profinite one $\widehat{GT}$ which does contain $G_{\mathbb{Q}}$ and a group $GT(k)$ defined for every field $k$. A deep result of Drinfeld assert that this latter group is in some sense a universal automorphism group of braided monoidal categories. Indeed, it acts on the set of Drinfeld associator with coefficients in $k$.

Now, there is also a morphism

$G_{\mathbb{Q}}\rightarrow GT(\mathbb{Q}_{\ell})$

for every prime numbre $\ell$. Hence the absolute galois group acts on each kind of object in which associators come up (assuming that it is in a situation where one can work over $\mathbb{Q}_{\ell}$). It leads to, I think, quite surprizing examples like action on finite type invariants of knots and links, on quantization functor of Lie bialgebras, and several other constructions arising in deformation/quantization theory as they are often related to Drinfeld associators.

There are some nice examples in knot theory and quantum algebra.

If $X$ is an algebraic variety over $\mathbb{Q}$, there is a canonical outer action $G_{\mathbb{Q}} \rightarrow Out(\hat{\pi}_1(X(\mathbb{C})))$ (which also exists for other fields than $\mathbb{Q}$ using the so-called algebraic fundamental group but I don't want to say something wrong about that). Roughly speaking, this is because finite covering of $X$ can be defined over $\bar{\mathbb{Q}}$, together with the relation between (regular) finite covering and finite quotients of the fundamental group. This action thus has the same origine as for dessin d'enfants.

Or particular interest is the study of this action in the case $X$ is the moduli space of algebraic curves of genus $g$ with $n$ marked points. This was suggested in Grothendieck's esquisse, and leads to the so-called Grothendieck-Teichmuller theory which gives a rather explicit description of a group which actually contains $G_{\mathbb{Q}}$. (see Cartographic group and flat stringy connection or Where is a good place to start learning about the Grothendieck-Teichmuller group?)

Actually, there are several flavours of the Grothendieck-Teichmuller group: a profinite one $\widehat{GT}$ which does contain $G_{\mathbb{Q}}$ and a group $GT(k)$ defined for every field $k$. A deep result of Drinfeld assert that this latter group is in some sense a universal automorphism group of braided monoidal categories. Indeed, it acts on the set of Drinfeld associator with coefficients in $k$.

Now, there is also a morphism

$G_{\mathbb{Q}}\rightarrow GT(\mathbb{Q}_{\ell})$

for every prime numbre $\ell$. Hence the absolute galois group acts on each kind of object in which associators come up (assuming that it is in a situation where one can work over $\mathbb{Q}_{\ell}$). It leads to, I think, quite surprizing examples like action on finite type invariants of knots and links, on quantization functor of Lie bialgebras, and several other constructions arising in deformation/quantization theory as they are often related to Drinfeld associators.