Return to Question

$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\GL{GL}$Fix a group $\Gamma$ and a positive integer $n$. Let $c(q):=\lvert\Hom(\Gamma, \GL_n(\mathbb{F}_q)\rvert$ denote the number of homomorphisms from $\Gamma$ to $\GL_n(\mathbb{F}_q)$.

Question A: is $c_n(q)$ a polynomial in $q$? I.e., does there exist a polynomial $P=P_\Gamma$ such that $c_n(q)=P(q)$ for all prime powers $q$?

This is known to be true if $\Gamma$ is finite or Fuchsian (which includes surface groups). For finite groups, see for instance the 2018 article "Asymptotic bounds for the size of $\Hom(A, \GL_n(q))$" by Bate and Gullon. For Fuchsian groups, this can be extracted from the discussions in the 2005 paper "Fuchsian groups, finite simple groups and representation varieties" by Liebeck and Shalev. As the latter reference makes clear, this is about counting points on the representation variety $\Hom(\Gamma, \GL_n)$. Thus, it can be considered as a question in arithmetic geomety.

Added in response to comments: the question should be rephrased as:

Question B: is $c_n(q)$ "essentially" a polynomial in $q$?

Here by "essentially" we could mean a number of things, e.g., excluding some primes, or allowing quasi-polynomials, etc.

Even a broader question:

Question C: What do we know about $c_n(q)$ for groups $\Gamma$ beyond the ones mentioned?

$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\GL{GL}$Fix a group $\Gamma$ and a positive integer $n$. Let $c(q):=\lvert\Hom(\Gamma, \GL_n(\mathbb{F}_q)\rvert$ denote the number of homomorphisms from $\Gamma$ to $\GL_n(\mathbb{F}_q)$.

Question A: is $c_n(q)$ a polynomial in $q$? I.e., does there exist a polynomial $P=P_\Gamma$ such that $c_n(q)=P(q)$ for all prime powers $q$?

This is known to be true if $\Gamma$ is finite or Fuchsian (which includes surface groups). For finite groups, see for instance the 2018 article "Asymptotic bounds for the size of $\Hom(A, \GL_n(q))$" by Bate and Gullon. For Fuchsian groups, this can be extracted from the discussions in the 2005 paper "Fuchsian groups, finite simple groups and representation varieties" by Liebeck and Shalev.

As the latter reference makes clear, (if $\Gamma$ is finitely generated), then this is about counting points on the representation variety $\Hom(\Gamma, \GL_n)$. Thus, it can be considered as a question in arithmetic geomety.

Added in response to comments: the question should be rephrased as:

Question B: is $c_n(q)$ "essentially" a polynomial in $q$?

Here by "essentially" we could mean a number of things, e.g., excluding some primes, or allowing quasi-polynomials, etc.

Even a broader question:

Question C: What do we know about $c_n(q)$ for groups $\Gamma$ beyond the ones mentioned?

$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\GL{GL}$Fix a group $\Gamma$ and a positive integer $n$. Let $c(q):=\lvert\Hom(\Gamma, \GL_n(\mathbb{F}_q)\rvert$ denote the number of homomorphisms from $\Gamma$ to $\GL_n(\mathbb{F}_q)$.

Question A: is $c_n(q)$ a polynomial in $q$? I.e., does there exist a polynomial $P=P_\Gamma$ such that $c_n(q)=P(q)$ for all prime powers $q$?

This is known to be true if $\Gamma$ is finite or Fuchsian (which includes surface groups). For finite groups, see for instance the 2018 article "Asymptotic bounds for the size of $\Hom(A, \GL_n(q))$" by Bate and Gullon. For Fuchsian groups, this can be extracted from the discussions in the 2005 paper "Fuchsian groups, finite simple groups and representation varieties" by Liebeck and Shalev. As the latter reference makes clear, this is about counting points on the representation variety $\Hom(\Gamma, \GL_n)$. Thus, it can be considered as a question in arithmetic geomety.

Added in edit: We can remove finitely many primes if necessary. We can also relax the requirement a bit and ask if the function is a quasi-polynomial or Polynomial on Residue Classes (PORC). The question can thus be rephrased as:

Question B: is $c_n(q)$ "essentially" a polynomial in $q$?

Even a broader question:

Question C: What do we know about $c_n(q)$ for groups $\Gamma$ beyond the ones mentioned?

$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\GL{GL}$Fix a group $\Gamma$ and a positive integer $n$. Let $c(q):=\lvert\Hom(\Gamma, \GL_n(\mathbb{F}_q)\rvert$ denote the number of homomorphisms from $\Gamma$ to $\GL_n(\mathbb{F}_q)$.

Question A: is $c_n(q)$ a polynomial in $q$? I.e., does there exist a polynomial $P=P_\Gamma$ such that $c_n(q)=P(q)$ for all prime powers $q$?

This is known to be true if $\Gamma$ is finite or Fuchsian (which includes surface groups). For finite groups, see for instance the 2018 article "Asymptotic bounds for the size of $\Hom(A, \GL_n(q))$" by Bate and Gullon. For Fuchsian groups, this can be extracted from the discussions in the 2005 paper "Fuchsian groups, finite simple groups and representation varieties" by Liebeck and Shalev. As the latter reference makes clear, this is about counting points on the representation variety $\Hom(\Gamma, \GL_n)$. Thus, it can be considered as a question in arithmetic geomety.

Added in response to comments: the question should be rephrased as:

Question B: is $c_n(q)$ "essentially" a polynomial in $q$?

Here by "essentially" we could mean a number of things, e.g., excluding some primes, or allowing quasi-polynomials, etc.

Even a broader question:

Question C: What do we know about $c_n(q)$ for groups $\Gamma$ beyond the ones mentioned?

Fix a group $\Gamma$ and a positive integer $n$. Let $c(q):=|\mathrm{Hom}(\Gamma, \mathrm{GL}_n(\mathbb{F}_q)|$ denote the number of homomorphisms from $\Gamma$ to $\mathrm{GL}_n(\mathbb{F}_q)$.

Question A: is $c_n(q)$ a polynomial in $q$? I.e., does there exist a polynomial $P=P_\Gamma$ such that $c_n(q)=P(q)$ for all prime powers $q$?

This is known to be true If $\Gamma$ is finite or Fuchsian (which includes surface groups). For finite groups, see for instance this 2018 article by Bate and Gullon. For Fuchsian groups, this can be extracted from the discussions in this 2005 paper by Liebeck and Shalev. As the latter reference makes clear, this is about counting points on the representation variety $\mathrm{Hom}(\Gamma, \mathrm{GL}_n)$. Thus, it can be considered as a question in arithmetic geomety.

Added in edit: We can remove finitely many primes if necessary. We can also relax the requirement a bit and ask if the function is a quasi-polynomial or Polynomial on Residue Classes (PORC). The question can thus be rephrased as:

Question B: is c_n(q) "essentially" a polynomial in q?

Even a more broader question:

Question C: What do we know about c_n(q) for groups $\Gamma$ beyond the ones mentioned?

$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\GL{GL}$Fix a group $\Gamma$ and a positive integer $n$. Let $c(q):=\lvert\Hom(\Gamma, \GL_n(\mathbb{F}_q)\rvert$ denote the number of homomorphisms from $\Gamma$ to $\GL_n(\mathbb{F}_q)$.

Question A: is $c_n(q)$ a polynomial in $q$? I.e., does there exist a polynomial $P=P_\Gamma$ such that $c_n(q)=P(q)$ for all prime powers $q$?

This is known to be true if $\Gamma$ is finite or Fuchsian (which includes surface groups). For finite groups, see for instance the 2018 article "Asymptotic bounds for the size of $\Hom(A, \GL_n(q))$" by Bate and Gullon. For Fuchsian groups, this can be extracted from the discussions in the 2005 paper "Fuchsian groups, finite simple groups and representation varieties" by Liebeck and Shalev. As the latter reference makes clear, this is about counting points on the representation variety $\Hom(\Gamma, \GL_n)$. Thus, it can be considered as a question in arithmetic geomety.

Added in edit: We can remove finitely many primes if necessary. We can also relax the requirement a bit and ask if the function is a quasi-polynomial or Polynomial on Residue Classes (PORC). The question can thus be rephrased as:

Question B: is $c_n(q)$ "essentially" a polynomial in $q$?

Even a broader question:

Question C: What do we know about $c_n(q)$ for groups $\Gamma$ beyond the ones mentioned?