  We study natural examples of binary channels with synchronization errors.
These include the duplication channel, which independently outputs a given bit
once or twice, and geometric channels that repeat a given bit according to a
geometric rule, with or without the possibility of bit deletion. We apply the
general framework of Cheraghchi (STOC 2018) to obtain sharp analytical upper
bounds on the capacity of these channels. Previously, upper bounds were known
via numerical computations involving the computation of finite approximations
of the channels by a computer and then using the obtained numerical results to
upper bound the actual capacity. While leading to sharp numerical results,
further progress on the full understanding of the channel capacity inherently
remains elusive using such methods. Our results can be regarded as a major step
towards a complete understanding of the capacity curves. Quantitatively, our
upper bounds sharply approach, and in some cases surpass, the bounds that were
previously only known by purely numerical methods. Among our results, we
notably give a completely analytical proof that, when the number of repetitions
per bit is geometric (supported on $\{0,1,2,\dots\}$) with mean growing to
infinity, the channel capacity remains substantially bounded away from $1$.
