How useful are geometric aspects when studying finite groups?












11














My newbie impression when studying finite group theory is that geometric aspects are not very prevalent. Cayley graphs play quite some role for visualising finite groups, but compared to the study of infinite groups in geometric group theory, geometric aspects seem to be marginal.



I therefore wonder:



(1) Why is geometry less interesting when studying finite groups (if this is really the case)?



(2) Could it (nevertheless) be interesting to delve into geometric group theory when studying finite groups?



I find GGT particularly interesting, but as I am currently working in finite group theory I am not sure how advisable it is to hope for connections. Finite groups are special cases of finitely generated infinite groups, but I am not sure if these are uninteresting cases from the GGT viewpoint.



Thank you for any clarifications and hints!






group-theory finite-groups geometric-group-theory







share|cite|improve this question


















  • 1




    I can't imagine how you reached the conclusion that I might mean that.
    – Chris Eagle
    Jun 19 '13 at 14:36






  • 1




    Bridson-Haefliger talk a lot about quasi-stuff. Quasi-isometry, quasi-convex, and so on. In finite groups, these just don't make sense - every finite group is quasi-isomorphic to the trivial group! (Because finite groups are virtually trivial...hehehe...) Thinking about trees (a la Meier) does not work because whenever a finite group acts on a tree there is a global fixed point.
    – user1729
    Jun 19 '13 at 14:39








  • 4




    However, have you ever come across "cyclically presented" groups? John Conway once, in, like, American Mathematical Monthly, asked if the group $langle a,b,c,d,e; ab=c, bc=d, cd=e, de=a, ea=brangle$ is finite or not. An apparently "fun" problem. The solutions took three years (it is cyclic of order 11, if you are wondering). It then became a research "theme" to work out when these groups with $n$ generators were finite, and Roger Lyndon (of Lyndon and Schupp), used small-cancellation theory to prove that they are infinite for $ngeq 11$. So geometric group theory!
    – user1729
    Jun 19 '13 at 14:43






  • 7




    Geometry for finite groups usually means “finite geometry” and “incidence geometry”. Part of it should have the same feel, but most of it will be pretty different. “Buildings”, “Coset geometries”, and “Subgroup complexes” are all ways that geometry works in finite groups.
    – Jack Schmidt
    Jun 19 '13 at 14:43






  • 2




    What the literature calls "geometric group theory" is more of an infinite groups thing. There are other geometric things going on in finite groups though, like symmetric genus, graphs of groups on surfaces, symmetries of finite geometries, etc.
    – Alexander Gruber♦
    Jun 20 '13 at 8:22


















11














My newbie impression when studying finite group theory is that geometric aspects are not very prevalent. Cayley graphs play quite some role for visualising finite groups, but compared to the study of infinite groups in geometric group theory, geometric aspects seem to be marginal.



I therefore wonder:



(1) Why is geometry less interesting when studying finite groups (if this is really the case)?



(2) Could it (nevertheless) be interesting to delve into geometric group theory when studying finite groups?



I find GGT particularly interesting, but as I am currently working in finite group theory I am not sure how advisable it is to hope for connections. Finite groups are special cases of finitely generated infinite groups, but I am not sure if these are uninteresting cases from the GGT viewpoint.



Thank you for any clarifications and hints!






group-theory finite-groups geometric-group-theory







share|cite|improve this question


















  • 1




    I can't imagine how you reached the conclusion that I might mean that.
    – Chris Eagle
    Jun 19 '13 at 14:36






  • 1




    Bridson-Haefliger talk a lot about quasi-stuff. Quasi-isometry, quasi-convex, and so on. In finite groups, these just don't make sense - every finite group is quasi-isomorphic to the trivial group! (Because finite groups are virtually trivial...hehehe...) Thinking about trees (a la Meier) does not work because whenever a finite group acts on a tree there is a global fixed point.
    – user1729
    Jun 19 '13 at 14:39








  • 4




    However, have you ever come across "cyclically presented" groups? John Conway once, in, like, American Mathematical Monthly, asked if the group $langle a,b,c,d,e; ab=c, bc=d, cd=e, de=a, ea=brangle$ is finite or not. An apparently "fun" problem. The solutions took three years (it is cyclic of order 11, if you are wondering). It then became a research "theme" to work out when these groups with $n$ generators were finite, and Roger Lyndon (of Lyndon and Schupp), used small-cancellation theory to prove that they are infinite for $ngeq 11$. So geometric group theory!
    – user1729
    Jun 19 '13 at 14:43






  • 7




    Geometry for finite groups usually means “finite geometry” and “incidence geometry”. Part of it should have the same feel, but most of it will be pretty different. “Buildings”, “Coset geometries”, and “Subgroup complexes” are all ways that geometry works in finite groups.
    – Jack Schmidt
    Jun 19 '13 at 14:43






  • 2




    What the literature calls "geometric group theory" is more of an infinite groups thing. There are other geometric things going on in finite groups though, like symmetric genus, graphs of groups on surfaces, symmetries of finite geometries, etc.
    – Alexander Gruber♦
    Jun 20 '13 at 8:22
















11












11








11


4





My newbie impression when studying finite group theory is that geometric aspects are not very prevalent. Cayley graphs play quite some role for visualising finite groups, but compared to the study of infinite groups in geometric group theory, geometric aspects seem to be marginal.



I therefore wonder:



(1) Why is geometry less interesting when studying finite groups (if this is really the case)?



(2) Could it (nevertheless) be interesting to delve into geometric group theory when studying finite groups?



I find GGT particularly interesting, but as I am currently working in finite group theory I am not sure how advisable it is to hope for connections. Finite groups are special cases of finitely generated infinite groups, but I am not sure if these are uninteresting cases from the GGT viewpoint.



Thank you for any clarifications and hints!






group-theory finite-groups geometric-group-theory







share|cite|improve this question













My newbie impression when studying finite group theory is that geometric aspects are not very prevalent. Cayley graphs play quite some role for visualising finite groups, but compared to the study of infinite groups in geometric group theory, geometric aspects seem to be marginal.



I therefore wonder:



(1) Why is geometry less interesting when studying finite groups (if this is really the case)?



(2) Could it (nevertheless) be interesting to delve into geometric group theory when studying finite groups?



I find GGT particularly interesting, but as I am currently working in finite group theory I am not sure how advisable it is to hope for connections. Finite groups are special cases of finitely generated infinite groups, but I am not sure if these are uninteresting cases from the GGT viewpoint.



Thank you for any clarifications and hints!






group-theory finite-groups geometric-group-theory




group-theory finite-groups geometric-group-theory






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Jun 19 '13 at 14:13









InfinitelyInquisitive

14419




14419








  • 1




    I can't imagine how you reached the conclusion that I might mean that.
    – Chris Eagle
    Jun 19 '13 at 14:36






  • 1




    Bridson-Haefliger talk a lot about quasi-stuff. Quasi-isometry, quasi-convex, and so on. In finite groups, these just don't make sense - every finite group is quasi-isomorphic to the trivial group! (Because finite groups are virtually trivial...hehehe...) Thinking about trees (a la Meier) does not work because whenever a finite group acts on a tree there is a global fixed point.
    – user1729
    Jun 19 '13 at 14:39








  • 4




    However, have you ever come across "cyclically presented" groups? John Conway once, in, like, American Mathematical Monthly, asked if the group $langle a,b,c,d,e; ab=c, bc=d, cd=e, de=a, ea=brangle$ is finite or not. An apparently "fun" problem. The solutions took three years (it is cyclic of order 11, if you are wondering). It then became a research "theme" to work out when these groups with $n$ generators were finite, and Roger Lyndon (of Lyndon and Schupp), used small-cancellation theory to prove that they are infinite for $ngeq 11$. So geometric group theory!
    – user1729
    Jun 19 '13 at 14:43






  • 7




    Geometry for finite groups usually means “finite geometry” and “incidence geometry”. Part of it should have the same feel, but most of it will be pretty different. “Buildings”, “Coset geometries”, and “Subgroup complexes” are all ways that geometry works in finite groups.
    – Jack Schmidt
    Jun 19 '13 at 14:43






  • 2




    What the literature calls "geometric group theory" is more of an infinite groups thing. There are other geometric things going on in finite groups though, like symmetric genus, graphs of groups on surfaces, symmetries of finite geometries, etc.
    – Alexander Gruber♦
    Jun 20 '13 at 8:22
















  • 1




    I can't imagine how you reached the conclusion that I might mean that.
    – Chris Eagle
    Jun 19 '13 at 14:36






  • 1




    Bridson-Haefliger talk a lot about quasi-stuff. Quasi-isometry, quasi-convex, and so on. In finite groups, these just don't make sense - every finite group is quasi-isomorphic to the trivial group! (Because finite groups are virtually trivial...hehehe...) Thinking about trees (a la Meier) does not work because whenever a finite group acts on a tree there is a global fixed point.
    – user1729
    Jun 19 '13 at 14:39








  • 4




    However, have you ever come across "cyclically presented" groups? John Conway once, in, like, American Mathematical Monthly, asked if the group $langle a,b,c,d,e; ab=c, bc=d, cd=e, de=a, ea=brangle$ is finite or not. An apparently "fun" problem. The solutions took three years (it is cyclic of order 11, if you are wondering). It then became a research "theme" to work out when these groups with $n$ generators were finite, and Roger Lyndon (of Lyndon and Schupp), used small-cancellation theory to prove that they are infinite for $ngeq 11$. So geometric group theory!
    – user1729
    Jun 19 '13 at 14:43






  • 7




    Geometry for finite groups usually means “finite geometry” and “incidence geometry”. Part of it should have the same feel, but most of it will be pretty different. “Buildings”, “Coset geometries”, and “Subgroup complexes” are all ways that geometry works in finite groups.
    – Jack Schmidt
    Jun 19 '13 at 14:43






  • 2




    What the literature calls "geometric group theory" is more of an infinite groups thing. There are other geometric things going on in finite groups though, like symmetric genus, graphs of groups on surfaces, symmetries of finite geometries, etc.
    – Alexander Gruber♦
    Jun 20 '13 at 8:22










1




1




I can't imagine how you reached the conclusion that I might mean that.
– Chris Eagle
Jun 19 '13 at 14:36




I can't imagine how you reached the conclusion that I might mean that.
– Chris Eagle
Jun 19 '13 at 14:36




1




1




Bridson-Haefliger talk a lot about quasi-stuff. Quasi-isometry, quasi-convex, and so on. In finite groups, these just don't make sense - every finite group is quasi-isomorphic to the trivial group! (Because finite groups are virtually trivial...hehehe...) Thinking about trees (a la Meier) does not work because whenever a finite group acts on a tree there is a global fixed point.
– user1729
Jun 19 '13 at 14:39






Bridson-Haefliger talk a lot about quasi-stuff. Quasi-isometry, quasi-convex, and so on. In finite groups, these just don't make sense - every finite group is quasi-isomorphic to the trivial group! (Because finite groups are virtually trivial...hehehe...) Thinking about trees (a la Meier) does not work because whenever a finite group acts on a tree there is a global fixed point.
– user1729
Jun 19 '13 at 14:39






4




4




However, have you ever come across "cyclically presented" groups? John Conway once, in, like, American Mathematical Monthly, asked if the group $langle a,b,c,d,e; ab=c, bc=d, cd=e, de=a, ea=brangle$ is finite or not. An apparently "fun" problem. The solutions took three years (it is cyclic of order 11, if you are wondering). It then became a research "theme" to work out when these groups with $n$ generators were finite, and Roger Lyndon (of Lyndon and Schupp), used small-cancellation theory to prove that they are infinite for $ngeq 11$. So geometric group theory!
– user1729
Jun 19 '13 at 14:43




However, have you ever come across "cyclically presented" groups? John Conway once, in, like, American Mathematical Monthly, asked if the group $langle a,b,c,d,e; ab=c, bc=d, cd=e, de=a, ea=brangle$ is finite or not. An apparently "fun" problem. The solutions took three years (it is cyclic of order 11, if you are wondering). It then became a research "theme" to work out when these groups with $n$ generators were finite, and Roger Lyndon (of Lyndon and Schupp), used small-cancellation theory to prove that they are infinite for $ngeq 11$. So geometric group theory!
– user1729
Jun 19 '13 at 14:43




7




7




Geometry for finite groups usually means “finite geometry” and “incidence geometry”. Part of it should have the same feel, but most of it will be pretty different. “Buildings”, “Coset geometries”, and “Subgroup complexes” are all ways that geometry works in finite groups.
– Jack Schmidt
Jun 19 '13 at 14:43




Geometry for finite groups usually means “finite geometry” and “incidence geometry”. Part of it should have the same feel, but most of it will be pretty different. “Buildings”, “Coset geometries”, and “Subgroup complexes” are all ways that geometry works in finite groups.
– Jack Schmidt
Jun 19 '13 at 14:43




2




2




What the literature calls "geometric group theory" is more of an infinite groups thing. There are other geometric things going on in finite groups though, like symmetric genus, graphs of groups on surfaces, symmetries of finite geometries, etc.
– Alexander Gruber♦
Jun 20 '13 at 8:22






What the literature calls "geometric group theory" is more of an infinite groups thing. There are other geometric things going on in finite groups though, like symmetric genus, graphs of groups on surfaces, symmetries of finite geometries, etc.
– Alexander Gruber♦
Jun 20 '13 at 8:22












1 Answer
1






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It depends what you mean by "geometric aspects". I think the place that your question comes from is that nobody uses the methods commonly referred to as "Geometric Group Theory" (such as those explained, for example, here [PDF of A Course in Geometric Group Theory, by Brian Bowditch), then the answer is on page 24: many of the main tools of Geometric Group Theory involve studying things up to quasi-isomorphism, and all finite groups are quasi-isomorphic to each other, so there isn't much to do.



As a secondary thing, the reason for your impression might be rather the opposite: it's not that geometric methods are less common in finite group theory, it's that other methods are more common. That is: a lot of the geometric stuff with infinite groups still works fine for finite groups, it's just rather drowned out by the significantly larger body of other tools that we can use.



However, there are very rich areas of group theory which do use geometric methods, interpreted more broadly. A handful have been mentioned in the comments, but I'll go for something else (mostly because it's what I know about):



Given a (finite) group $G$, if we want to study its representations, we can often have a problem of getting our hands on those representations in a form that's amenable to further study (pick any finite group whose irreducible representations are not thoroughly known, and this is likely at play there somewhere). So it can be useful to have ways to generate representations of a given group, which you can then work with.



One reasonably common way to do this is to take an action of $G$ on something that has a homology theory (for our purposes, some geometric structure $X$), compute the homology of $X$ (in the normal topological sense, with some suitable coefficients), and pass the action of $G$ on $X$ through to get an action of $G$ on the homology terms, which are modules, and therefore give you representations of $G$. As a few examples: you can take the building of $G$, which gives some reasonably well-understood representations, you can take the simplicial complex of the poset of the subgroups of $G$, which gives you some somewhat less well-understood groups (but give me a bit on that, I'm working on it!), or you can take some complex derived from some natural action of your group (for example, the action of $S_n$ on a set of $n$ points gives an action on the set of collections of (proper) disjoint subsets of $S_n$, which form a simplicial complex with the subsets as vertices in the obvious way, which gives you some nice representations), or you can take the subgroup complexes mentioned in the comments.






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    It depends what you mean by "geometric aspects". I think the place that your question comes from is that nobody uses the methods commonly referred to as "Geometric Group Theory" (such as those explained, for example, here [PDF of A Course in Geometric Group Theory, by Brian Bowditch), then the answer is on page 24: many of the main tools of Geometric Group Theory involve studying things up to quasi-isomorphism, and all finite groups are quasi-isomorphic to each other, so there isn't much to do.



    As a secondary thing, the reason for your impression might be rather the opposite: it's not that geometric methods are less common in finite group theory, it's that other methods are more common. That is: a lot of the geometric stuff with infinite groups still works fine for finite groups, it's just rather drowned out by the significantly larger body of other tools that we can use.



    However, there are very rich areas of group theory which do use geometric methods, interpreted more broadly. A handful have been mentioned in the comments, but I'll go for something else (mostly because it's what I know about):



    Given a (finite) group $G$, if we want to study its representations, we can often have a problem of getting our hands on those representations in a form that's amenable to further study (pick any finite group whose irreducible representations are not thoroughly known, and this is likely at play there somewhere). So it can be useful to have ways to generate representations of a given group, which you can then work with.



    One reasonably common way to do this is to take an action of $G$ on something that has a homology theory (for our purposes, some geometric structure $X$), compute the homology of $X$ (in the normal topological sense, with some suitable coefficients), and pass the action of $G$ on $X$ through to get an action of $G$ on the homology terms, which are modules, and therefore give you representations of $G$. As a few examples: you can take the building of $G$, which gives some reasonably well-understood representations, you can take the simplicial complex of the poset of the subgroups of $G$, which gives you some somewhat less well-understood groups (but give me a bit on that, I'm working on it!), or you can take some complex derived from some natural action of your group (for example, the action of $S_n$ on a set of $n$ points gives an action on the set of collections of (proper) disjoint subsets of $S_n$, which form a simplicial complex with the subsets as vertices in the obvious way, which gives you some nice representations), or you can take the subgroup complexes mentioned in the comments.






    share|cite|improve this answer


























      0














      It depends what you mean by "geometric aspects". I think the place that your question comes from is that nobody uses the methods commonly referred to as "Geometric Group Theory" (such as those explained, for example, here [PDF of A Course in Geometric Group Theory, by Brian Bowditch), then the answer is on page 24: many of the main tools of Geometric Group Theory involve studying things up to quasi-isomorphism, and all finite groups are quasi-isomorphic to each other, so there isn't much to do.



      As a secondary thing, the reason for your impression might be rather the opposite: it's not that geometric methods are less common in finite group theory, it's that other methods are more common. That is: a lot of the geometric stuff with infinite groups still works fine for finite groups, it's just rather drowned out by the significantly larger body of other tools that we can use.



      However, there are very rich areas of group theory which do use geometric methods, interpreted more broadly. A handful have been mentioned in the comments, but I'll go for something else (mostly because it's what I know about):



      Given a (finite) group $G$, if we want to study its representations, we can often have a problem of getting our hands on those representations in a form that's amenable to further study (pick any finite group whose irreducible representations are not thoroughly known, and this is likely at play there somewhere). So it can be useful to have ways to generate representations of a given group, which you can then work with.



      One reasonably common way to do this is to take an action of $G$ on something that has a homology theory (for our purposes, some geometric structure $X$), compute the homology of $X$ (in the normal topological sense, with some suitable coefficients), and pass the action of $G$ on $X$ through to get an action of $G$ on the homology terms, which are modules, and therefore give you representations of $G$. As a few examples: you can take the building of $G$, which gives some reasonably well-understood representations, you can take the simplicial complex of the poset of the subgroups of $G$, which gives you some somewhat less well-understood groups (but give me a bit on that, I'm working on it!), or you can take some complex derived from some natural action of your group (for example, the action of $S_n$ on a set of $n$ points gives an action on the set of collections of (proper) disjoint subsets of $S_n$, which form a simplicial complex with the subsets as vertices in the obvious way, which gives you some nice representations), or you can take the subgroup complexes mentioned in the comments.






      share|cite|improve this answer
























        0












        0








        0






        It depends what you mean by "geometric aspects". I think the place that your question comes from is that nobody uses the methods commonly referred to as "Geometric Group Theory" (such as those explained, for example, here [PDF of A Course in Geometric Group Theory, by Brian Bowditch), then the answer is on page 24: many of the main tools of Geometric Group Theory involve studying things up to quasi-isomorphism, and all finite groups are quasi-isomorphic to each other, so there isn't much to do.



        As a secondary thing, the reason for your impression might be rather the opposite: it's not that geometric methods are less common in finite group theory, it's that other methods are more common. That is: a lot of the geometric stuff with infinite groups still works fine for finite groups, it's just rather drowned out by the significantly larger body of other tools that we can use.



        However, there are very rich areas of group theory which do use geometric methods, interpreted more broadly. A handful have been mentioned in the comments, but I'll go for something else (mostly because it's what I know about):



        Given a (finite) group $G$, if we want to study its representations, we can often have a problem of getting our hands on those representations in a form that's amenable to further study (pick any finite group whose irreducible representations are not thoroughly known, and this is likely at play there somewhere). So it can be useful to have ways to generate representations of a given group, which you can then work with.



        One reasonably common way to do this is to take an action of $G$ on something that has a homology theory (for our purposes, some geometric structure $X$), compute the homology of $X$ (in the normal topological sense, with some suitable coefficients), and pass the action of $G$ on $X$ through to get an action of $G$ on the homology terms, which are modules, and therefore give you representations of $G$. As a few examples: you can take the building of $G$, which gives some reasonably well-understood representations, you can take the simplicial complex of the poset of the subgroups of $G$, which gives you some somewhat less well-understood groups (but give me a bit on that, I'm working on it!), or you can take some complex derived from some natural action of your group (for example, the action of $S_n$ on a set of $n$ points gives an action on the set of collections of (proper) disjoint subsets of $S_n$, which form a simplicial complex with the subsets as vertices in the obvious way, which gives you some nice representations), or you can take the subgroup complexes mentioned in the comments.






        share|cite|improve this answer












        It depends what you mean by "geometric aspects". I think the place that your question comes from is that nobody uses the methods commonly referred to as "Geometric Group Theory" (such as those explained, for example, here [PDF of A Course in Geometric Group Theory, by Brian Bowditch), then the answer is on page 24: many of the main tools of Geometric Group Theory involve studying things up to quasi-isomorphism, and all finite groups are quasi-isomorphic to each other, so there isn't much to do.



        As a secondary thing, the reason for your impression might be rather the opposite: it's not that geometric methods are less common in finite group theory, it's that other methods are more common. That is: a lot of the geometric stuff with infinite groups still works fine for finite groups, it's just rather drowned out by the significantly larger body of other tools that we can use.



        However, there are very rich areas of group theory which do use geometric methods, interpreted more broadly. A handful have been mentioned in the comments, but I'll go for something else (mostly because it's what I know about):



        Given a (finite) group $G$, if we want to study its representations, we can often have a problem of getting our hands on those representations in a form that's amenable to further study (pick any finite group whose irreducible representations are not thoroughly known, and this is likely at play there somewhere). So it can be useful to have ways to generate representations of a given group, which you can then work with.



        One reasonably common way to do this is to take an action of $G$ on something that has a homology theory (for our purposes, some geometric structure $X$), compute the homology of $X$ (in the normal topological sense, with some suitable coefficients), and pass the action of $G$ on $X$ through to get an action of $G$ on the homology terms, which are modules, and therefore give you representations of $G$. As a few examples: you can take the building of $G$, which gives some reasonably well-understood representations, you can take the simplicial complex of the poset of the subgroups of $G$, which gives you some somewhat less well-understood groups (but give me a bit on that, I'm working on it!), or you can take some complex derived from some natural action of your group (for example, the action of $S_n$ on a set of $n$ points gives an action on the set of collections of (proper) disjoint subsets of $S_n$, which form a simplicial complex with the subsets as vertices in the obvious way, which gives you some nice representations), or you can take the subgroup complexes mentioned in the comments.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Dec 3 '18 at 13:09









        user3482749

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            Sphinx de Gizeh

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            Pour les articles homonymes, voir Sphinx. Sphinx de Gizeh.mw-parser-output .entete.egypte-antique{background-image:url("//upload.wikimedia.org/wikipedia/commons/c/cc/Picto_infobox_ancient_Egypt.png")} Site d'Égypte antique Le visage du sphinx de Gizeh. Noms en arabe أبو الهول , Abou al-Hôl Localisation Région Basse-Égypte Coordonnées 29° 58′ 31″ nord, 31° 08′ 16″ est Géolocalisation sur la carte : Égypte Sphinx de Gizeh modifier   Le sphinx de Gizeh est la statue thérianthrope qui se dresse devant les grandes pyramides du plateau de Gizeh, en Basse-Égypte. Sculpture monumentale monolithique la plus grande du monde [ 1 ] avec 73,5 mètres de longueur, 14 mètres de largeur et 20,22 mètres de hauteur, elle représente un sphinx couchant. Réalisée vers -2500, elle serait attribuée à Djédefrê, l'un des pharaons de la IV e  dynastie, représentant son père le pharaon Khéops. Sommaire 1 Étymologie 2 Desc...
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            Dijon

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            Get global maximum slope

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            up vote 1 down vote favorite I have a function which computes values on images with the size of 32x32 pixels. Therefore, the function is applied on every single pixel $x_i$ but also depends on all 32x32 input pixels. The input is constrained to $-1 leq x_i leq 1$ . More formal: A function $f_i:mathbb [-1,1]^{1024} rightarrow mathbb R$ and an input vector $vec{x} in mathbb [-1,1]^{1024}$ . Now I am looking for the global maximum slope of $f_i$ (for any input) numerically. A local maximum slope is easy to find as one goes in the direction of the gradient. How to find the global maximum? I am not sure if brute force is an option because when the pixel values have a high precision (e.g. 1e-10) then the total number of input permutations is ridiculously high. The brute force method might work with a lower precision but this doesn...
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