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If you fix $N$ and group elements $g_1$, ..., $g_N \in G$, then your question becomes closely related to tilings of groups. Specifically, in Chapter 2, section 2 of "Entropy and Isomorphism Theorems for Actions of Amenable Groups," Ornstein and Weiss prove:

Let $G$ be a countable group acting freely and measure preservingly on a standard probability space $(X, \mu)$. Fix a finite set $T \subseteq G$. If for every $\epsilon > 0$ there is a measurable set $U \subseteq X$ such that the $T$-translates of $U$ are disjoint and $\mu(T \cdot U) > 1 - \epsilon$, then $T$ tiles $G$ in the sense that there is a set of centers $C \subseteq G$ such that the sets $Tc$ ($c \in C$) partition $G$.

They also prove that if $G$ is amenable then the converse holds. So if $G$ is amenable and $T$ is a tile for $G$, then for every free probability measure preserving action of $G$ and every $\epsilon > 0$ there is a $(\epsilon, |T|)$-fundamentalish domain. Thus a natural question is: which amenable groups admit arbitrarily large finite tiles?

Weiss called a group $G$ MT (mono-tileable) if for every finite set $F \subseteq G$ there is a finite tile $T \subseteq G$ containing $F$. In "Monotileable amenable groups," Weiss proved that all solvable groups and all residually finite groups are MT. In "Elementary Amenable Groups," Chou proved that all elementary amenable groups and all free products of non-trivial groups are MT. So in particular, your question has a positive answer whenever the group $G$ is elementary amenable. A stronger tiling condition, called ccc, is studied in chapter 4 of "Groups Colorings and Bernoulli Subflows" (this paper is in preparation). Weaker properties of poly-MT and poly-ccc are studied in "Burnside's Problem, spanning trees, and tilings." To the best of my knowledge these are the only papers which study tilings of countable groups.

If you fix $N$ and group elements $g_1$, ..., $g_N \in G$, then your question becomes closely related to tilings of groups. Specifically, in Chapter 2, section 2 of "Entropy and Isomorphism Theorems for Actions of Amenable Groups," Ornstein and Weiss prove:

Let $G$ be a countable group acting freely and measure preservingly on a standard probability space $(X, \mu)$. Fix a finite set $T \subseteq G$. If for every $\epsilon > 0$ there is a measurable set $U \subseteq X$ such that the $T$-translates of $U$ are disjoint and $\mu(T \cdot U) > 1 - \epsilon$, then $T$ tiles $G$ in the sense that there is a set of centers $C \subseteq G$ such that the sets $Tc$ ($c \in C$) partition $G$.

They also prove that if $G$ is amenable then the converse holds. So if $G$ is amenable and $T$ is a tile for $G$, then for every free probability measure preserving action of $G$ and every $\epsilon > 0$ there is a $(\epsilon, |T|)$-fundamentalish domain. Thus a natural question is: which amenable groups admit arbitrarily large finite tiles?

Weiss called a group $G$ MT (mono-tileable) if for every finite set $F \subseteq G$ there is a finite tile $T \subseteq G$ containing $F$. In "Monotileable amenable groups," Weiss proved that all solvable groups and all residually finite groups are MT. In "Elementary Amenable Groups," Chou proved that all elementary amenable groups and all free products of non-trivial groups are MT. So in particular, your question has a positive answer whenever the group $G$ is elementary amenable. A stronger tiling condition, called ccc, is studied in chapter 4 of "Groups Colorings and Bernoulli Subflows" (this paper is in preparation). Weaker properties of poly-MT and poly-ccc are studied in "Burnside's Problem, spanning trees, and tilings." To the best of my knowledge these are the only papers which study tilings of countable groups.

If you fix $N$ and group elements $g_1$, ..., $g_N \in G$, then your question becomes closely related to tilings of groups. Specifically, in Chapter 2, section 2 of "Entropy and Isomorphism Theorems for Actions of Amenable Groups," Ornstein and Weiss prove:

Let $G$ be a countable group acting freely and measure preservingly on a standard probability space $(X, \mu)$. Fix a finite set $T \subseteq G$. If for every $\epsilon > 0$ there is a measurable set $U \subseteq X$ such that the $T$-translates of $U$ are disjoint and $\mu(T \cdot U) > 1 - \epsilon$, then $T$ tiles $G$ in the sense that there is a set of centers $C \subseteq G$ such that the sets $Tc$ ($c \in C$) partition $G$.

They also prove that if $G$ is amenable then the converse holds. So if $G$ is amenable and $T$ is a tile for $G$, then for every free probability measure preserving action of $G$ and every $\epsilon > 0$ there is a $(\epsilon, |T|)$-fundamentalish domain. Thus a natural question is: which amenable groups admit arbitrarily large finite tiles?

Weiss called a group $G$ MT (mono-tileable) if for every finite set $F \subseteq G$ there is a finite tile $T \subseteq G$ containing $F$. In "Monotileable amenable groups," Weiss proved that all solvable groups and all residually finite groups are MT. In "Elementary Amenable Groups," Chou proved that all elementary amenable groups and all free products of non-trivial groups are MT. So in particular, your question has a positive answer whenever the group $G$ is elementary amenable. A stronger tiling condition, called ccc, is studied in chapter 4 of "Groups Colorings and Bernoulli Subflows" (this paper is in preparation). Weaker properties of poly-MT and poly-ccc are studied in "Burnside's Problem, spanning trees, and tilings." To the best of my knowledge these are the only papers which study tilings of countable groups.

If you fix $N$ and group elements $g_1$, ..., $g_N \in G$, then your question becomes closely related to tilings of groups. Specifically, in Chapter 2, section 2 of "Entropy and Isomorphism Theorems for Actions of Amenable Groups," Ornstein and Weiss prove:

Let $G$ be a countable group acting freely and measure preservingly on a standard probability space $(X, \mu)$. Fix a finite set $T \subseteq G$. If for every $\epsilon > 0$ there is a measurable set $U \subseteq X$ such that the $T$-translates of $U$ are disjoint and $\mu(T \cdot U) > 1 - \epsilon$, then $T$ tiles $G$ in the sense that there is a set of centers $C \subseteq G$ such that the sets $Tc$ ($c \in C$) partition $G$.

They also prove that if $G$ is amenable then the converse holds. So if $G$ is amenable and $T$ is a tile for $G$, then for every free probability measure preserving action of $G$ and every $\epsilon > 0$ there is a $(\epsilon, |T|)$-fundamentalish domain. Thus a natural question is: which amenable groups admit arbitrarily large finite tiles?

Weiss called a group $G$ MT (mono-tileable) if for every finite set $F \subseteq G$ there is a finite tile $T \subseteq G$ containing $F$. In "Monotileable amenable groups," Weiss proved that all solvable groups and all residually finite groups are MT. In "Elementary Amenable Groups," Chou proved that all elementary amenable groups and all free products of non-trivial groups are MT. So in particular, your question has a positive answer whenever the group $G$ is elementary amenable. A stronger tiling condition, called ccc, is studied in chapter 4 of "Groups Colorings and Bernoulli Subflows" (this paper is in preparation). Weaker properties of poly-MT and poly-ccc are studied in "Burnside's Problem, spanning trees, and tilings." To the best of my knowledge these are the only papers which study tilings of countable groups.

If you fix $N$ and group elements $g_1$, ..., $g_N \in G$, then your question becomes closely related to tilings of groups. Specifically, in Chapter 2, section 2 of "Entropy and Isomorphism Theorems for Actions of Amenable Groups," Ornstein and Weiss prove:

Let $G$ be a countable group acting freely and measure preservingly on a standard probability space $(X, \mu)$. Fix a finite set $T \subseteq G$. If for every $\epsilon > 0$ there is a measurable set $U \subseteq X$ such that the $T$-translates of $U$ are disjoint and $\mu(T \cdot U) > 1 - \epsilon$, then $T$ tiles $G$ in the sense that there is a set of centers $C \subseteq G$ such that the sets $Tc$ ($c \in C$) partition $G$.

They also prove that if $G$ is amenable then the converse holds. So if $G$ is amenable and $T$ is a tile for $G$, then for every free probability measure preserving action of $G$ and every $\epsilon > 0$ there is a $(\epsilon, |T|)$-fundamentalish domain. Thus a natural question is: which amenable groups admit arbitrarily large finite tiles?

Weiss called a group $G$ MT (mono-tileable) if for every finite set $F \subseteq G$ there is a finite tile $T \subseteq G$ containing $F$. In "Monotileable amenable groups," Weiss proved that all solvable groups and all residually finite groups are MT. In "Elementary Amenable Groups," Chou proved that all elementary amenable groups and all free products of non-trivial groups are MT. So in particular, your question has a positive answer whenever the group $G$ is elementary amenable. A stronger tiling condition, called ccc, is studied in chapter 4 of "Groups Colorings and Bernoulli Subflows" (this paper is in preparation). Weaker properties of poly-MT and poly-ccc are studied in "Burnside's Problem, spanning trees, and tilings." To the best of my knowledge these are the only papers which study tilings of countable groups.

If you fix $N$ and group elements $g_1$, ..., $g_N \in G$, then your question becomes closely related to tilings of groups. Specifically, in Chapter 2, section 2 of "Entropy and Isomorphism Theorems for Actions of Amenable Groups," Ornstein and Weiss prove:

Let $G$ be a countable group acting freely and measure preservingly on a standard probability space $(X, \mu)$. Fix a finite set $T \subseteq G$. If for every $\epsilon > 0$ there is a measurable set $U \subseteq X$ such that the $T$-translates of $U$ are disjoint and $\mu(T \cdot U) > 1 - \epsilon$, then $T$ tiles $G$ in the sense that there is a set of centers $C \subseteq G$ such that the sets $Tc$ ($c \in C$) partition $G$.

They also prove that if $G$ is amenable then the converse holds. So if $G$ is amenable and $T$ is a tile for $G$, then for every free probability measure preserving action of $G$ and every $\epsilon > 0$ there is a $(\epsilon, |T|)$-fundamentalish domain. Thus a natural question is: which amenable groups admit arbitrarily large finite tiles?

Weiss called a group $G$ MT (mono-tileable) if for every finite set $F \subseteq G$ there is a finite tile $T \subseteq G$ containing $F$. In "Monotileable amenable groups," Weiss proved that all solvable groups and all residually finite groups are MT. In "Elementary Amenable Groups," Chou proved that all elementary amenable groups and all free products of non-trivial groups are MT. So in particular, your question has a positive answer whenever the group $G$ is elementary amenable. A stronger tiling condition, called ccc, is studied in chapter 4 of "Groups Colorings and Bernoulli Subflows" (this paper is in preparation). Weaker properties of poly-MT and poly-ccc are studied in "Burnside's Problem, spanning trees, and tilings." To the best of my knowledge these are the only papers which study tilings of countable groups.