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Register a new userOn $k$-free-like groups
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A $k$-free like group is a $k$-generated group $G$ with a sequence of $k$-element generating sets $Z_n$ such that the girth of $G$ relative to $Z_n$ is unbounded and the Cheeger constant of $G$ relative to $Z_n$ is bounded away from 0. By a recent result of Benjamini-Nachmias-Peres, this implies that the critical bond percolation probability of the Cayley graph of $G$ relative to $Z_n$ tends to $1/(2k-1)$ as $nto infty$. Answering a question of Benjamini, we construct many non-free groups that are $k$-free like for all sufficiently large $k$.
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For $G$ a finitely generated group and $g in G$, we say $g$ is detected by a normal subgroup $N lhd G$ if $g otin N$. The depth $D_G(g)$ of $g$ is the lowest index of a normal, finite index subgroup $N$ that detects $g$. In this paper we study the expected depth, $mathbb E[D_G(X_n)]$, where $X_n$ is a random walk on $G$. We give several criteria that imply that $$mathbb E[D_G(X_n)] xrightarrow[nto infty]{} 2 + sum_{k geq 2}frac{1}{[G:Lambda_k]}, ,$$ where $Lambda_k$ is the intersection of all normal subgroups of index at most $k$. In particular, the equality holds in the class of all nilpotent groups and in the class of all linear groups satisfying Kazhdan Property $(T)$. We explain how the right-hand side above appears as a natural limit and also give an example where the convergence does not hold.
We consider finite sums of counting functions on the free group $F_n$ and the free monoid $M_n$ for $n geq 2$. Two such sums are considered equivalent if they differ by a bounded function. We find the complete set of linear relations between equivalence classes of sums of counting functions and apply this result to construct an explicit basis for the vector space of such equivalence classes. Moreover, we provide a graphical algorithm to determine whether two given sums of counting functions are equivalent. In particular, this yields an algorithm to decide whether two sums of Brooks quasimorphisms on $F_n$ represent the same class in bounded cohomology.
We classify the boundaries of hyperbolic groups that have enough quasiconvex codimension-1 surface subgroups with trivial or cyclic intersections.
Let $F$ be a free group of finite rank. We say that the monomorphism problem in $F$ is decidable if for any two elements $u$ and $v$ in $F$, there is an algorithm that determines whether there exists a monomorphism of $F$ that sends $u$ to $v$. In this paper we show that the monomorphism problem is decidable and we provide an effective algorithm that solves the problem.
We build two non-abelian CSA-groups in which maximal abelian subgroups are conjugate and divisible.
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